m  w  I 


iliiii 


il'll'*;!' 


Q/^29 


\iin\     iii 


mmmm    auh 


Digitized  by  the  Internet  Archive 

in  2010  with  funding  from 

Boston  Library  Consortium  IVIember  Libraries 


http://www.archive.org/details/williamoughtredgOOcajo 


WILLIAM  OUGHTRED 


WILLIAM  OUGHTRED 

A    GREAT    SEVENTEENTH-CENTURY 
TEACHER    OF 

MATHEMATICS 


BY 

FLORIAN  CAJORI,  Ph.D. 

professor  of  Mathematics 
Colorado  College 


CHICAGO  LONDON 

THE    OPEN   COURT   PUBLISHING   COMPANY 

1916 


Copyright  1916  By 
The  Open  Court  Publishing  Co. 


All  Rights  Reserved 


Published  September  1916 


Composed  and  Printed  By 

The  University  of  Chicago  Press 

Chicago,  Illinois,  U.S.A. 


150408 


TABLE  OF  CONTENTS 

PAGE 

Introduction i 

CHAPTER 

I.  Oughtred's  Life 3 

At  School  and  University 3 

As  Rector  and  Amateur  Mathematician  ...  6 

His  Wife 7 

In  Danger  of  Sequestration 8 

His  Teaching 9 

Appearance  and  Habits 12 

Alleged  Travel  Abroad 14 

His  Death 15 

II.  Principal  Works 17 

Clavis  mathematicae 17 

Circles  of  Proportion  and  Trigonometrie    .      .      .35 

Solution  of  Numerical  Equations 39 

Logarithms 46 

Invention  of   the   Slide  Rule;   Controversy  on 

Priority  of  Invention 46 

III.  Minor  Works 50 

IV.  Oughtred's  Influence  upon  Mathematical  Prog- 

ress AND  Teaching 57 

Oughtred  and  Harriot 57 

Oughtred's  Pupils 58 

Oughtred,  the  "Todhunter  of  the  Seventeenth 

Century" 60 

Was  Descartes  Indebted  to  Oughtred  ?    .      .      .69 

The  Spread  of  Oughtred's  Notations  ....  73 

V 


vi  Table  of  Contents 

CHAPTER  PAGE 

V.  Oughtred's  Ideas  on  the  Teaching  oe  Mathe- 
matics   84 

General  Statement 84 

Mathematics,  "a  Science  of  the  Eye"      ...  85 

Rigorous  Thinking  and  the  Use  of  Instruments  .  87 

Newton's  Comments  on  Oughtred      ....  94 

Index 97 


INTRODUCTION 

In  the  year  1660  the  Royal  Society  was  founded  by 
royal  favor  in  London,  although  in  reality  its  inception 
took  place  in  1645  when  the  Philosophical  Society  (or, 
as  Boyle  called  it,  the  "Invisible  College")  came  into 
being,  which  held  meetings  at  Gresham  College  in  London 
and  later  in  Oxford.  It  was  during  the  second  half  of  the 
seventeenth  century  that  Sir  Isaac  Newton,  surrounded 
by  a  group  of  great  men — Wallis,  Hooke,  Barrow,  Halley, 
Cotes — carried  on  his  epoch-making  researches  in  mathe- 
matics, astronomy,  and  physics.  But  it  is  not  this  half- 
century  of  science  in  England,  nor  any  of  its  great  men,  that 
especially  engage  our  attention  in  this  monograph.  It  is 
rather  the  half-century  preceding,  an  epoch  of  prepara- 
tion, when  in  the  early  times  of  the  House  of  Stuart  the 
sciences  began  to  flourish  in  England.  Says  Dr.  A.  E. 
Shipley:  "Whatever  were  the  political  and  moral  de- 
ficiencies of  the  Stuart  kings,  no  one  of  them  lacked  intel- 
ligence in  things  artistic  and  scientific."  It  was  at  this 
time  that  mathematics,  and  particularly  algebra,  began 
to  be  cultivated  with  greater  zeal,  when  elementary  alge- 
bra with  its  symbolism  as  we  know  it  now  began  to  take 
its  shape. 

Biographers  of  Sir  Isaac  Newton  make  particular  men- 
tion of  five  mathematical  books  which  he  read  while  a 
young  student  at  Cambridge,  namely,  Euclid's  Elements, 
Descartes's  Geometrie,  Vieta's  Works,  Van  Schooten's 
Miscellanies,  and  Oughtred's  Clavis  mathematicae.  The 
last  of  these  books  has  been  receiving  increasing  attention 


2  William  Oughtred 

from  the  historians  of  algebra  in  recent  years.  We  have 
prepared  this  sketch  because  we  felt  that  there  were  points 
of  interest  in  the  Ufe  and  activity  of  Oughtred  which  have 
not  received  adequate  treatment.  Historians  have  dis- 
cussed his  share  in  the  development  of  symbolic  algebra, 
but  some  have  fallen  into  errors,  due  to  inability  to 
examine  the  original  editions  of  Oughtred's  Clavis  mathe- 
maticae,  which  are  quite  rare  and  inaccessible  to  most 
readers.  Moreover,  historians  have  failed  utterly  to 
recognize  his  inventions  of  mathematical  instruments, 
particularly  the  slide  rule;  they  have  completely  over- 
looked his  educational  views  and  his  ideas  on  mathematical 
teaching.  The  modern  reader  may  pause  with  profit 
to  consider  briefly  the  career  of  this  interesting  man. 

Oughtred  was  not  a  professional  mathematician.  He 
did  not  make  his  livelihood  as  a  teacher  of  mathematics 
or  as  a  writer,  nor  as  an  engineer  who  applies  mathematics 
to  the  control  and  use  of  nature's  forces.  Oughtred  was 
by  profession  a  minister  of  the  gospel.  With  him  the 
study  of  mathematics  was  a  side  issue,  a  pleasure,  a  recrea- 
tion. Like  the  great  French  algebraist,  Vieta,  from  whom 
he  drew  much  of  his  inspiration,  he  was  an  amateur  mathe- 
matician. The  word  "amateur"  must  not  be  taken  here 
in  the  sense  of  superficial  or  unthorough.  Great  Britain 
has  had  many  men  distinguished  in  science  who  pursued 
science  as  amateurs.     Of  such  men  Oughtred  is  one  of 

the  very  earliest. 

F.  C. 


CHAPTER  I 

OUGHTRED'S  LIFE 

AT   SCHOOL  AND  UNIVERSITY 

William  Oughtred,  or,  as  he  sometimes  wrote  his  name, 
Owtred,  was  born  at  Eton,  the  seat  of  Eton  College,  the 
year  of  his  birth  being  variously  given  as  1573,  1574,  and 
1575.  "His  father,"  says  Aubrey,  "taught  to  write  at 
Eaton,  and  was  a  scrivener;  and  understood  common 
arithmetique,  and  'twas  no  small  helpe  and  furtherance 
to  his  son  to  be  instructed  in  it  when  a  schoole-boy."^ 
He  was  a  boy  at  Eton  in  the  year  of  the  Spanish  Armada. 
At  this  famous  school,  which  prepared  boys  for  the  uni- 
versities, young  Oughtred  received  thorough  training  in 
classical  learning. 

According  to  information  received  from  F.  L.  Clarke, 
Bursar  and  Clerk  of  King's  College,  Cambridge,  Oughtred 
was  admitted  at  King's  a  scholar  from  Eton  on  Septem- 
ber I,  1592,  at  the  age  of  seventeen.  He  was  made  Fellow 
at  King's  on  September  i,  1595,  while  Elizabeth  was  still 
on  the  throne.  He  received  in  1596  the  degree  of  Bachelor 
of  Arts  and  in  1600  that  of  Master  of  Arts.  He  vacated 
his  fellowship  about  the  beginning  of  August,  1603.  His 
career  at  the  University  of  Cambridge  we  present  in  his 
own  words.     He  says: 

Next  after  Eaton  schoole,  I  was  bred  up  in  Cambridge  in 
Kings  Colledge :  of  which  society  I  was  a  member  about  eleven 
or  twelve  yeares :  wherein  how  I  behaved  my  self e,  going  hand 
in  hand  with  the  rest  of  my  ranke  in  the  ordinary  Academicall 

^  Aubrey's  Brief  Lives,  ed.  A.  Clark,  Vol.  II,  Oxford,  1898,  p.  106. 

3 


4  William  Oughtred 

studies  and  exercises,  and  with  what  approbation,  is  well 
knowne  and  remembered  by  many:  the  time  which  over  and 
above  those  usuall  studies  I  employed  upon  the  Mathematical! 
sciences,  I  redeemed  night  by  night  from  my  naturall  sleep, 
defrauding  my  body,  and  inuring  it  to  watching,  cold,  and 
labour,  while  most  others  tooke  their  rest.  Neither  did  I 
therein  seek  only  my  private  content,  but  the  benefit  of  many: 
and  by  inciting,  assisting,  and  instructing  others,  brought 
many  into  the  love  and  study  of  those  Arts,  not  only  in  our 
own,  but  in  some  other  Colledges  also:  which  some  at  this  time 
(men  far  better  than  my  selfe  in  learning,  degree,  and  prefer- 
ment) will  most  lovingly  acknowledge.^ 

These  words  describe  the  struggles  which  every  youth 
not  endowed  with  the  highest  genius  must  make  to  achieve 
success.  They  show,  moreover,  the  kindly  feeling  toward 
others  and  the  delight  he  took  throughout  life  in  assisting 
anyone  interested  in  mathematics.  Oughtred's  passion 
for  this  study  is  the  more  remarkable  as  neither  at  Eton 
nor  at  Cambridge  did  it  receive  emphasis.  Even  after 
his  time  at  Cambridge  mathematical  studies  and  their 
applications  were  neglected  there.  Jeremiah  Horrox 
was  at  Cambridge  in  1633-35,  desiring  to  make  himself 
an  astronomer. 

"But  many  impediments,"  says  Horrox,  "presented  them- 
selves: the  tedious  difficulty  of  the  study  itself  deterred  a  mind 
not  yet  formed;  the  want  of  means  oppressed,  and  still  op- 
presses, the  aspirations  of  my  mind:   but  that  which  gave  me 

^  "To  the  English  Gentrie,  and  all  others  studious  of  the  Mathe- 
maticks,  which  shall  bee  Readers  hereof.  The  just  Apologie  of  Wil: 
Ovghtred,  against  the  slaunderous  insimulations  of  Richard  Dela- 
main,  in  a  Pamphlet  called  Grammelogia,  or  the  Mathematicall  Ring, 
or  Mirifica  logarithmorum  projectio  circularis"  [1633?],  p.  8.  Here- 
after we  shall  refer  to  this  pamphlet  as  the  Apologeticall  Epistle,  this 
name  appearing  on  the  page-headings. 


Oughtred^s  Life  5 

most  concern  was  that  there  was  no  one  who  could  instruct 
me  in  the  art,  who  could  even  help  my  endeavours  by  joining 
me  in  the  study;   such  was  the  sloth  and  languor  which  had 

seized  all I  found  that  books  must  be  used  instead  of 

teachers."^ 

Some  attention  was  given  to  Greek  mathematicians, 
but  the  v/orks  of  Italian,  German,  and  French  algebraists 
of  the  latter  part  of  the  sixteenth  and  beginning  of  the 
seventeenth  century  were  quite  unknown  at  Cambridge 
in  Oughtred's  day.  It  was  part  of  his  life-work  as  a 
mathematician  to  make  algebra,  as  it  was  being  developed 
in  his  time,  accessible  to  English  youths. 

At  the  age  of  twenty-three  Oughtred  invented  his 
Easy  Way  of  Delineating  Sun- Dials  by  Geometry,  which, 
though  not  published  until  about  half  a  century  later, 
in  the  first  English  edition  of  Oughtred's  Clavis  mathe- 
maticae  in  1647,  was  in  the  meantime  translated  into 
Latin  by  Christopher  Wren,  then  a  Gentleman  Com- 
moner of  Wadham  College,  Oxford,  now  best  known 
through  his  architectural  creations.  In  1600  Oughtred 
wrote  a  monograph  on  the  construction  of  sun-dials 
upon  a  plane  of  any  inclination,  but  that  paper  was 
withheld  by  him  from  publication  until  1632.  Sun- 
dials were  interesting  objects  of  study,  since  watches 
and  pendulum  clocks  were  then  still  unknown.  All  sorts 
of  sun-dials,  portable  and  non-portable,  were  used  at 
that  time  and  long  afterward.  Several  of  the  college 
buildings  at  Oxford  and  Cambridge  have  sun-dials  even 
at  the  present  time. 

^  Companion  to  the  [British]  Almanac  of  1837,  p.  28,  in  an  article 
by  i\.ugustus  De  Morgan  on  "Notices  of  English  Mathematical  and 
Astronomical  Writers  between  the  Norman  Conquest  and  the  Year 
1600." 


6  William  Oughtred 

AS   RECTOR   AND   AMATEUR   MATHEMATICAN 

It  was  in  1604  that  Oughtred  entered  upon  his  profes- 
sional Ufe-wbrk  as  a  preacher,  being  instituted  to  the 
vicarage  of  Shalford  in  Surrey.  In  1610  he  was  made 
rector  of  Albury,  where  he  spent  the  remainder  of  his  long 
life.  Since  the  era  of  the  Reformation  two  of  the  rectors 
of  Albury  obtained  great  celebrity  from  their  varied  talents 
and  acquirements — our  William  Oughtred  and  Samuel 
Horsley.  Oughtred  continued  to  devote  his  spare  time 
to  mathematics,  as  he  had  done  in  college.  A  great  mathe- 
matical invention  made  by  a  Scotchman  soon  commanded 
his  attention — the  invention  of  logarithms.  An  informant 
writes  as  follows: 

Lord  Napier,  in  1614,  published  at  Edinburgh  his  Mirifici 

logarithmorum  canonis  descriptio It  presently  fell  into 

the  hands  of  Mr.  Briggs,  then  geometry-reader  at  Gresham 
College  in  London:  and  that  gentleman,  forming  a  design  to 
perfect  Lord  Napier's  plan,  consulted  Oughtred  upon  it;  who 
probably  wrote  his  Treatise  of  Trigonometry  about  the  same 
time,  since  it  is  evidently  formed  upon  the  plan  of  Lord  Napier's 
CanonJ 

It  will  be  shown  later  that  Oughtred  is  very  probably 
the  author  of  an  "Appendix"  which  appeared  in  the  16 18 
edition  of  Edward  Wright's  translation  into  English  of 
John  Napier's  Descriptio.  This  "Appendix"  relates  to 
logarithms  and  is  an  able  document,  containing  several 
points  of  historical  interest.  Mr.  Arthur  Hutchinson  of 
Pembroke  College  informs  me  that  in  the  university 
library  at  Cambridge  there  is  a  copy  of  Napier's  Con- 
strudio  (16 19)  bound  up  with  a  copy  of  Kepler's  Chilias 
logarithmorum  (1624),  that  at  the  beginning  of  the  Con- 

^  New  and  General  Biographical  Dictionary  (John  Nichols),  Lon- 
don, 1784,  art.  "Oughtred." 


Oughtred^s  Life  7 

structio  is  a  blank  leaf,  and  before  this  occurs  the  title- 
page  only  of  Napier's  Descriptio  (16 19),  at  the  top  of 
which  appears  Oughtred's  autograph.  The  history  of 
this  interesting  signature  is  unknown. 

HIS   WIFE 

In  1606  he  married  Christ 'sgift  Gary  11,  daughter  of 
Caryll,  Esq.,  of  Tangley,  in  an  adjoining  parish.^  We 
know  very  little  about  Oughtred's  family  life.  The 
records  at  King's  College,  Cambridge,^  mention  a  son, 
but  it  is  certain  that  there  were  more  children.  A 
daughter  was  married  to  Christopher  Brookes.  But 
there  is  no  confirmation  of  Aubrey's  statements,^  accord- 
ing to  which  Oughtred  had  nine  sons  and  four  daughters. 
Reference  to  the  wife  and  children  is  sometimes  made  in 
the  correspondence  with  Oughtred.  In  16 16  J.  Hales 
writes,  "I  pray  let  me  be  remembered,  though  unknown, 
to  Mistress  Oughtred. "^ 

As  we  shall  see  later,  Oughtred  had  a  great  many  young 
men  who  came  to  his  house  and  remained  there  free  of 
charge  to  receive  instruction  in  mathematics,  which  was 
likewise  gratuitous.  This  being  the  case,  certainly  great 
appreciation  was  due  to  Mrs.  Oughtred,  upon  whom  the 
burden  of  hospitality  must  have  fallen.  Yet  chroniclers 
are  singularly  silent  in  regard  to  her.  Hers  was  evidently 
a  life  of  obscurity  and  service.     We  greatly  doubt  the 

^  Rev.  Owen  Manning,  History  of  Antiquities  in  Surrey,  Vol.  II, 
p.  132. 

2  Skeleton  CoUegii  Regalis  Cantab.:  Or  A  Catalogue  of  All  the 
Provosts,  Fellows  and  Scholars,  of  the  King's  College  ....  since  the 
Foundation  Thereof,  Vol.  II,  "WiUiam  Oughtred." 

3  Aubrey,  op.  cit.,  Vol.  II,  p.  107. 

4  Rigaud,  Correspondence  of  Scientific  Men  of  the  Seventeenth 
Century,  Oxford,  Vol.  I,  1841,  p.  5. 


8  William  Oughtred 

accuracy  of  the  following  item  handed  down  by  Aubrey; 
it  cannot  be  a  true  characterization: 

His  wife  was  a  penurious  woman,  and  would  not  allow  him 
to  burne  candle  after  supper,  by  which  meanes  many  a  good 
notion  is  lost,  and  many  a  probleme  unsolved;  so  that  Mr. 
[Thomas]  Henshawe,  when  he  was  there,  bought  candle,  which 
was  a  great  comfort  to  the  old  man.^ 

IN  DANGER   OF   SEQUESTRATION 

Oughtred  spent  his  years  in  ''unremitted  attention  to 
his  favourite  study,"  sometimes,  it  has  been  whispered,  to 
the  neglect  of  his  rectorial  duties.     Says  Aubrey: 

I  have  heard  his  neighbour  ministers  say  that  he  was  a 
pittiful  preacher;  the  reason  was  because  he  never  studyed 
it,  but  bent  all  his  thoughts  on  the  mathematiques;  but  when 
he  was  in  danger  of  being  sequestred  for  a  royahst,  he  fell  to 
the  study  of  divinity,  and  preacht  (they  sayd)  admirably 
weU,  even  in  his  old  age.^ 

This  remark  on  sequestration  brings  to  mind  one  of  the 
political  and  religious  struggles  of  the  time,  the  episcopacy 
against  the  independent  movements.     Says  Manning: 

In  1646  he  was  cited  before  the  Committee  for  Ecclesiastical 
Affairs,  where  many  articles  had  been  deposed  against  him; 
but,  by  the  favour  of  Sir  Bulstrode  Whitlock  and  others,  who, 
at  the  intercession  of  William  Lilye  the  Astrologer,  appeared 
in  great  numbers  on  his  behalf,  he  had  a  majority  on  his  side, 
and  so  escaped  a  sequestration.^ 

Not  without  interest  is  the  account  of  this  matter  given 
by  Lilly  himself : 

About  this  Time,  the  most  famous  Mathematician  of  all 
Europe,  (Mr.  William  Oughtred,  Parson  of  Aldbury  in  Surrey) 

^  Aubrey,  op.  cii.,  Vol.  II,  p.  no. 

^Ibid.,  p.  III.  3  op.  cit.,  Vol.  II,  p.  132. 


Oughtred^s  Life  g 

was  in  Danger  of  Sequestration  by  the  Committee  of  or  for 
plunder 'd  Ministers;  (Ambo-dexters  they  were;)  several  incon- 
siderable Articles  were  deposed  and  sworn  against  him,  material 
enough  to  have  sequestred  him,  but  that,  upon  his  Day  of 
hearing,  I  applied  my  self  to  Sir  Bolstrode  Whitlock,  and  all  my 
own  old  Friends,  who  in  such  Numbers  appeared  in  his  Behalf, 
that  though  the  Chairman  and  many  other  Presbyterian 
Members  were  stiff  against  him,  yet  he  was  cleared  by  the 
major  Number.  The  truth  is,  he  had  a  considerable  Parsonage, 
and  that  only  was  enough  to  sequester  any  moderate  Judgment : 
He  was  also  well  known  to  affect  his  Majesty  [Charles  I].  In 
these  Times  many  worthy  Ministers  lost  their  Livings  or  Bene- 
fices, for  not  complying  with  the  Three-penny  Directory.^ 

HIS   TEACHING 

Oughtred  had  few  personal  enemies.  His  pupils  held 
him  in  highest  esteem  and  showed  deep  gratitude;  only 
one  pupil  must  be  excepted,  Richard  Delamain.  Against 
him  arose  a  bitter  controversy  which  saddened  the  life 
of  Oughtred,  then  an  old  man.  It  involved,  as  we  shall 
see  later,  the  priority  of  invention  of  the  circular  slide 
rule  and  of  a  horizontal  instrument  or  portable  sun-dial. 
In  defense  of  himself,  Oughtred  wrote  in  1633  or  1634  the 
Apologeticall  Epistle,  from  which  we  quoted  above.  This 
document  contains  biographical  details,  in  part  as  follows : 

Ever  since  my  departure  from  the  Vniversity,  which  is 
about  thirty  yeares,  I  have  lived  neere  to  the  Towne  of  Guild- 
ford in  Surrey:  where,  whether  /  have  taken  so  much  liberty 
to  the  losse  of  time,  and  the  neglect  of  my  calling  the  whole  Coun- 
trey  thereabout,  both  Gentry  and  others,  to  whom  I  am  full 
well  knowne,  will  quickely  informe  him;  my  house  being  not 
past  three  and  twenty  miles  from  London:  and  yet  I  so  hid 
my  selve  at  home,  that  I  seldomly  travelled  so  farre  as  London 

^  Mr.  William  Lilly's  History  of  His  Life  and  Times,  From  the 
Year  1602  to  1681,  London,  1715,  p.  58. 


lo  William  Oughtred 

once  in  a  yeare.  Indeed  the  life  and  mind  of  man  cannot 
endure  without  some  interchangeablenesse  of  recreation,  and 
pawses  from  the  intensive  actions  of  our  severall  callings;  and 
every  man  is  drawne  with  his  owne  delight.  My  recreations 
have  been  diversity  of  studies:  and  as  oft  as  I  was  toyled  with 
the  labour  of  my  owne  profession,  I  have  allayed  that  tedious- 
nesse  by  walking  in  the  pleasant  and  more  then  Elysian  fields 
of  the  diverse  and  various  parts  of  humane  learning,  and  not 
the  Mathematics  onely. 

Even  the  opponents  of  Delamain  must  be  grateful 
to  him  for  having  been  the  means  of  drawing  from  Ought- 
red  such  interesting  biographical  details.  Oughtred 
proceeds  to  tell  how,  about  1628,  he  was  induced  to  write 
his  Clavis  mathematicae,  upon  which  his  reputation  as  a 
mathematician  largely  rests: 

About  five  yeares  smce,  the  Earle  of  ArundeU  my  most 
honourable  Lord  in  a  time  of  his  private  retiring  to  his  house 
in  the  coxmtrey  then  at  West  Horsley,  foure  small  miles  from 
me  (though  since  he  hath  a  house  in  Aldebury  the  parish  where 
I  live)  hearing  of  me  (by  what  meanes  I  know  not)  was  pleased 
to  send  for  me:  and  afterward  at  London  to  appoint  mee  a 
Chamber  of  his  owne  house:  where,  at  such  times,  and  in  such 
manner  as  it  seemed  him  good  to  imploy  me,  and  when  I 
might  not  inconveniently  be  spared  from  my  charge,  I  have 
been  most  ready  to  present  my  selfe  in  all  humble  and  affec- 
tionate service:  I  hope  also  without  the  offence  of  God,  the 
transgression  of  the  good  Lawes  of  this  Land,  neglect  of 
my  caUing,  or  the  deserved  scandall  of  any  good  man 

And  although  I  am  no  mercenary  man,  nor  make  profession 
to  teach  any  one  in  these  arts  for  gaine  and  recompence,  but 
as  I  serve  at  the  Altar,  so  I  live  onely  of  the  Altar:  yet  in  those 
interims  that  I  am  at  London  in  my  Lords  service,  I  have  been 
still  much  frequented  both  by  Natives  and  Strangers,  for  my 
resolution  and  instruction  in  many  difficult  poynts  of  Art; 
and  have  most  freely  and  lovingly  imparted  my  selfe  and  my 


Oughtred^s  Life  ii 

skill,  such  as  I  had,  to  their  contentments,  and  much  honourable 
acknowledgement  of  their  obligation  to  my  Lord  for  bringing 
mee  to  London,  hath  beene  testifyed  by  many.  Of  which  my 
HberaUity  and  unwearyed  readinesse  to  doe  good  to  all,  scarce 
any  one  can  give  more  ample  testimony  then  R.  D.  himselfe 
can:  would  he  be  but  pleased  to  allay  the  shame  of  this  his 
hot  and  eager  contention,  blowne  up  onely  with  the  full 
bellowes  of  intended  glory  and  gaine;  ....  they  [the  subjects 
in  which  Delamain  received  assistance  from  Oughtred]  were 
the  first  elements  of  Astronomie  concerning  the  second  motions 
of  the  fixed  starres,  and  of  the  Sunne  and  Moone;  they  were 
the  first  elements  of  Conies,  to  dehneate  those  sections:  they 
were  the  first  elements  of  Optics,  Catoptrics,  and  Dioptrics: 
of  all  which  you  knew  nothing  at  all. 

These  last  passages  are  instructive  as  showing  what 
topics  were  taken  up  for  study  with  some  of  his  pupils. 
The  chief  subject  of  interest  with  most  of  them  was  algebra, 
which  at  that  time  was  just  beginning  to  draw  the  atten- 
tion of  English  lovers  of  mathematics. 

Oughtred  carried  on  an  extensive  correspondence  on 
mathematical  subjects.  He  was  frequently  called  upon 
to  assist  in  the  solution  of  knotty  problems — sometimes 
to  his  annoyance,  perhaps,  as  is  shown  by  the  following 
letter  which  he  wrote  in  1642  to  a  stranger,  named 
Price : 

It  is  true  that  I  have  bestowed  such  vacant  time,  as  I  could 
gain  from  the  study  of  divinity,  (which  is  my  calUng,)  upon 
human  knowledges,  and,  amongst  other,  upon  the  mathematics, 
wherein  the  Httle  skill  I  have  attained,  being  compared  with 
others  of  my  profession,  who  for  the  most  part  contenting 
themselves  only  with  their  own  way,  refuse  to  tread  these  sale- 
brous  and  uneasy  paths,  may  peradventure  seem  the  more. 
But  now  being  in  years  and  mindful  of  mine  end,  and  having 
paid  dearly  for  my  former  delights  both  in  my  health  and  state, 


12  William  Oughtred 

besides  the  prejudice  of  such,  who  not  considering  what  inces- 
sant labour  may  produce,  reckon  so  much  wanting  unto  me  in 
my  proper  calling,  as  they  think  I  have  acquired  in  other 
sciences;  by  which  opinion  (not  of  the  vulgar  only)  I  have 
suffered  both  disrespect,  and  also  hinderance  in  some  small 
perferments  I  have  aimed  at.  I  have  therefore  now  learned 
to  spare  myself,  and  am  not  willing  to  descend  again  in  arenam, 
and  to  serve  such  ungrateful  muses.    Yet,  sir,  at  your  request 

I  have  perused  your  problem Your  problem  is  easily 

wrought  per  Nicomedis  conchoidem  lineam.^ 

APPEARANCE  AND  HABITS 

Aubrey  gives  information  about  the  appearance  and 
habits  of  Oughtred: 

He  w^as  a  little  man,  had  black  haire,  and  blacke  eies  (with  a 
great  deal  of  spirit) .  His  head  was  always  working.  He  would 
drawe  lines  and  diagrams  on  the  dust 

He  [his  oldest  son  Benjamin]  told  me  that  his  father  did  use 
to  lye  a  bed  till  eleaven  or  twelve  a  clock,  with  his  doublet  on, 
ever  since  he  can  remember.  Studyed  late  at  night;  went  not 
to  bed  till  1 1  a  clock;  had  his  tinder  box  by  him;  and  on  the  top 
of  his  bed-staffe,  he  had  his  inke-horne  fix't.  He  slept  but  little. 
Sometimes  he  went  not  to  bed  in  two  or  three  nights,  and  would 
not  come  downe  to  meales  till  he  had  found  out  the  quaesitiim. 

He  was  more  famous  abroad  for  his  learning,  and  more 
esteemed,  then  at  home.  Severall  great  mathematicians  came 
over  into  England  on  purpose  to  converse  with  him.  His 
countrey  neighbours  (though  they  understood  not  his  worth) 
knew  that  there  must  be  extraordinary  worth  in  him,  that  he 
was  so  visited  by  foreigners 

When  learned  foreigners  came  and  sawe  how  privately  he 
lived,  they  did  admire  and  blesse  themselves,  that  a  person  of 
so  much  worth  and  learning  should  not  be  better  provided 
for 

^  Rigaud,  op,  cit.,  Vol.  I,  p.  60. 


Oughtred^s  Life  J3 

He  has  told  bishop  Ward,  and  Mr.  Elias  Ashmole  (who  was 
his  neighbour),  that  "on  this  spott  of  ground"  (or  "leaning 
against  this  oake"  or  "that  ashe"),  "the  solution  of  such  or 
such  a  probleme  came  into  my  head,  as  if  infused  by  a  divine 
genius,  after  I  had  thought  on  it  without  successe  for  a  yeare, 
two,  or  three."  .... 

Nicolaus  Mercator,  Holsatus  ....  went  to  see  him  few 
yeares  before  he  dyed 

The  right  hon^^®  Thomas  Howard,  earle  of  Arundel  and 
Surrey,  Lord  High  Marshall  of  England,  was  his  great  patron, 
and  loved  him  intirely.  One  time  they  were  like  to  have  been 
killed  together  by  the  fall  at  Albury  of  a  grott,  which  fell 
downe  but  just  as  they  were  come  out.^ 

Oughtred's  friends  convey  the  impression  that,  in  the 
main,  Oughtred  enjoyed  a  comfortable  living  at  Albury. 
Only  once  appear  indications  of  financial  embarrassment. 
About  1634  one  of  his  pupils,  W.  Robinson,  writes  as 
follows: 

1  protest  unto  you  sincerely,  were  I  as  able  as  some,  at  whose 
hands  you  have  merited  exceedingly,  or  (to  speak  more  abso- 
lutely) as  able  as  willing,  I  would  as  freely  give  you  500  /. 
per  ann.  as  500  pence;  and  I  cannot  but  be  astonished  at  this 
our  age,  wherein  pelf  and  dross  is  made  their  summum  bonum, 
and  the  best  part  of  man,  with  the  true  ornaments  thereof, 
science  and  knowledge,  are  so  slighted ^ 

In  his  letters  Oughtred  complains  several  times  of  the 
limitations  for  work  and  the  infirmities  due  to  his  advan- 
cing old  age.  The  impression  he  made  upon  others  was 
quite  different.     Says  one  biographer: 

He  sometimes  amused  himself  with  archery,  and  sometimes 

practised  as  a  surveyor  of  land He  was  sprightly  and 

active,  when  more  than  eighty  years  of  age.^ 

^  Aubrey,  op,  cit.,  Vol.  II,  p.  107. 

2  Rigaud,  op.  cit.,  Vol.  I,  p.  16.         3  Owen  Manning,  op.  cit.,  p.  132. 


14  William  Oughtred 

Another  informant  says  that  Oughtred  was 

as  facetious  in  Greek  and  Latine  as  solid  in  Arithmetique, 
Astronomy,  and  the  sphere  of  all  Measures,  Musick,  etc. ;  exact 
in  his  style  as  in  his  judgment;  handling  his  Cube,  and  other 
Instruments  at  eighty,  as  steadily,  as  others  did  at  thirty; 
owing  this,  he  said,  to  temperance  and  Archery;  principling 
his  people  with  plain  and  soUd  truths,  as  he  did  the  world  with 
great  and  useful  Arts;  advancing  new  Inventions  in  all  things 
but  ReHgion.  Which  in  its  old  order  and  decency  he  main- 
tained secure  in  his  privacy,  prudence,  meekness,  simplicity, 
resolution,  patience,  and  contentment.^ 

ALLEGED  TRAVEL  ABROAD 

According  to  certain  sources  of  information,  Oughtred 
traveled  on  the  European  Continent  and  was  invited  to 
change  his  abode  to  the  Continent.  We  have  seen  no 
statement  from  Oughtred  himself  on  this  matter.  He 
seldom  referred  to  himself  in  his  books  and  letters.  The 
autobiography  contained  in  his  Apologeticall  Epistle 
was  written  a  quarter  of  a  century  before  his  death. 
Aubrey  gives  the  following: 

In  the  time  of  the  civill  warres  the  duke  of  Florence  invited 
him  over,  and  offered  him  500  H.  per  annum;  but  he  would  not 
accept  it,  because  of  his  religion.^ 

A  portrait  of  Oughtred,  painted  in  1646  by  Hollar  and 
inserted  in  the  English  edition  of  the  Clavis  of  1647,  con- 
tains underneath  the  following  lines: 

"Haec  est  Oughtredi  senio  labantis  imago 
Itala  quam  cupiit.  Terra  Britanna  tuHt." 

In  the  sketch  of  Oughtred  by  Owen  Manning  it  is 
confessed  that  "it  is  not  known  to  what  this  alludes;  but 

^  New  and  General  Biographical  Dictionary  (John  Nichols), 
London,  1784,  art.  "Oughtred." 

'  Op.  ciL,Yo\.  II,  p.  no. 


Oughtred's  Life  1 5 

possibly  he  might  have  been  in  Italy  with  his  patron,  the 
Earl  of  Arundel."^  It  would  seem  quite  certain  either 
that  Oughtred  traveled  in  Europe  or  that  he  received 
some  sort  of  an  offer  to  settle  in  Italy.  In  view  of  Aubrey's 
explicit  statement  and  of  Oughtred's  well-known  habit  of 
confining  himself  to  his  duties  and  studies  in  his  own 
parish,  seldom  going  even  as  far  as  London,  we  strongly 
incline  to  the  opinion  that  he  did  not  travel  on  the  Con- 
tinent, but  that  he  received  an  offer  from  some  patron 
of  the  sciences — ^possibly  some  distinguished  visitor — to 
settle  in  Italy. 

HIS   DEATH 

He  died  at  Albury,  June  30,  1660,  aged  about  eighty- 
six  years.  Of  his  last  days  and  death,  Aubrey  speaks  as 
follows: 

Before  he  dyed  he  burned  a  world  of  papers,  and  sayd  that 
the  world  was  not  worthy  of  them;  he  was  so  superb.  He 
burned  also  severall  printed  bookes,  and  would  not  stirre,  till 

they  were   consumed I    myselfe   have    his    Pitiscus, 

imbelished  with  his  excellent  marginaU  notes,  which  I  esteeme 
as  a  great  rarity.  I  wish  I  could  also  have  got  his  BUingsley's 
EucUd,  which  John  Collins  sayes  was  full  of  his  annota- 
tions  

Ralph  Greatrex,  his  great  friend,  the  mathematical! 
instrument-maker,  sayed  he  conceived  he  dyed  with  joy  for 
the  comeing-in  of  the  king,  which  was  the  29th  of  May  before. 
''And  are  yee  sure  he  is  restored?" — "Then  give  me  a  glasse 
of  sack  to  drinke  his  sacred  majestic 's  health."  His  spirits 
were  then  quite  upon  the  wing  to  fly  away ^ 

In  this  passage,  as  in  others,  due  allowance  must  be 
made  for  Aubrey's  lack  of  discrimination.     He  was  not 

^  Rev.  Owen  Manning,  The  History  and  Antiquities  of  Surrey, 
Vol.  II,  London,  1809,  p.  132. 

'  Op.  cit.,  Vol.  II,  1898,  p.  III. 


1 6  William  Oughtred 

in  the  habit  of  sifting  facts  from  mere  gossip.  That 
Oughtred  should  have  declared  that  the  world  was  not 
worthy  of  his  papers  or  manuscripts  is  not  in  consonance 
with  the  sweetness  of  disposition  ordinarily  attributed  to 
him.  More  probable  was  the  feeling  that  the  papers  he 
burned — possibly  old  sermons — were  of  no  particular 
value  to  the  world.  That  he  did  not  destroy  a  large  mass 
of  mathematical  manuscripts  is  evident  from  the  fact  that 
a  considerable  number  of  them  came  after  his  death  into 
the  hands  of  Sir  Charles  Scarborough,  M.D.,  under  whose 
supervision  some  of  them  were  carefully  revised  and  pub- 
lished at  Oxford  in  1677  under  the  title  oi  Opuscula  mathe- 
matica  hactenus  inedita. 

Aubrey's  story  of  Oughtred's  mode  of  death  has  been 
as  widely  circulated  in  every  modern  biographical  sketch 
as  has  his  slander  of  Mrs.  Oughtred  by  claiming  that  she 
was  so  penurious  that  she  would  deny  him  the  use  of 
candles  to  read  by.  Oughtred  died  on  June  30;  the  Res- 
toration occurred  on  May  29.  No  doubt  Oughtred 
rejoiced  over  the  Restoration,  but  the  story  of  his  drinking 
"a  glass  of  sack"  to  his  Majesty's  health,  and  then  dying 
of  joy  is  surely  apocryphal.  De  Morgan  humorously 
remarks,  "It  should  be  added,  by  way  of  excuse,  that  he 
was  eighty-six  years  old."^ 

^  Budget  of  Paradoxes,  London,  1872,  p.  451;  2d  ed.,  Chicago  and 
London,  19 15,  Vol.  II,  p.  303. 


CHAPTER  II 
PRINCIPAL  WORKS 


CLAVIS   MATHEMATICAE  " 


Passing  to  the  consideration  of  Oughtred's  mathe- 
matical books,  we  begin  with  the  observation  that  he 
showed  a  marked  disinclination  to  give  his  writings  to  the 
press.  His  first  paper  on  sun-dials  was  written  at  the  age 
of  twenty-three,  but  we  are  not  aware  that  more  than  one 
brief  mathematical  manuscript  was  printed  before  his 
fifty-seventh  year.  In  every  instance,  publication  in 
printed  form  seems  to  have  been  due  to  pressure  exerted 
by  one  or  more  of  his  patrons,  pupils,  or  friends.  Some 
of  his  manuscripts  were  lent  out  to  his  pupils,  who  prepared 
copies  for  their  own  use.  In  some  instances  they  urged 
upon  him  the  desirability  of  publication  and  assisted  in 
preparing  copy  for  the  printer.  The  earliest  and  best- 
known  book  of  Oughtred  was  his  Clavis  mathematicae, 
to  which  repeated  allusion  has  already  been  made.  As 
he  himself  informs  us,  he  was  employed  by  the  Earl  of 
Arundel  about  1628  to  instruct  the  Earl's  son,  Lord 
William  Howard  (afterward  Viscount  Stafford)  in  the 
mathematics.  For  the  use  of  this  young  man  Oughtred 
composed  a  treatise  on  algebra  which  was  published  in 
Latin  in  the  year  1631  at  the  urgent  request  of  a  kinsman 
of  the  young  man,  Charles  Cavendish,  a  patron  of  learning. 

The  Clavis  mathematicae,^  in  its  first  edition  of  163 1,  was 
a  booklet  of  only  88  small  pages.     Yet  it  contained  in  very 

^  The  full  tide  of  the  Clavis  of  1631  is  as  follows:  Arithmeticae 
in  nunieris  et  speciebvs  institvlio:   Qvae  tvm  logisticae,  tvm  analyticae, 

17 


1 8  William  Oughtred 

condensed  form  the  essentials  of  arithmetic  and  algebra  as 
known  at  that  time. 

Aside  from  the  addition  of  four  tracts,  the  163 1  edition 
underwent  some  changes  in  the  editions  of  1647  and  1648, 
which  two  are  much  alike.  The  twenty  chapters  of  1631 
are  reduced  to  nineteen  in  1647  ^^^  in  all  the  later  editions. 
Numerous  minute  alterations  from  the  163 1  edition  occur 
in  all  parts  of  the  books  of  1647  and  1648.  The  material 
of  the  last  three  chapters  of  the  163 1  edition  is  rearranged, 
with  some  slight  additions  here  and  there.  The  1648 
edition  has  no  preface.  In  the  print  of  1652  there  are  only 
slight  alterations  from  the  1648  edition;    after  that  the 

atgve  adeo  totivs  mathematicae,  qvasi  clavis  est. — Ad  nohilissimvm 
spectatissimumque  invenem  Dn.  Gvilelmvm  Howard,  Ordinis  qui  dicitur, 
Balnei  Eqiiitem,  honor atissimi  Dn.  Thomae,  Comitis  Arvndeliae  &" 
Svrriae,  Comitis  Mareschalli  Angliae,  b'c  filium. — Londini,  A  pud 
Thomam  Harpervm.     M.DC.XXXI. 

In  all  there  appeared  five  Latin  editions,  the  second  in  1648  at  Lon- 
don, the  third  in  1652  at  Oxford,  the  fourth  in  1667  at  Oxford,  the  fifth 
in  1693  and  1698  at  Oxford.  There  were  two  independent  English 
editions:  the  first  in  1647  at  London,  translated  in  greater  part  by- 
Robert  Wood  of  Lincoln  College,  Oxford,  as  is  stated  in  the  preface 
to  the  1652  Latin  edition;  the  second  in  1694  and  1702  is  a  new  trans- 
lation, the  preface  being  written  and  the  book  recommended  by  the 
astronomer  Edmund  Halley.  The  1694  and  1702  impressions  labored 
under  the  defect  of  many  sense-disturbing  errors  due  to  careless 
reading  of  the  proofs.  All  the  editions  of  the  Clavis,  after  the  first 
edition,  had  one  or  more  of  the  following  tracts  added  on : 

Eq.  =  De  Aequationum  afectarvm  resolvtione  in  numeris. 
Eu.=Ele?nenti  decimi  Euclidis  declaratio. 
So.  =  De  Solidis  regularibus,  tractatus. 
An.  =  De  Anatocismo,  sive  usura  composita. 
Fa.  =  Regula  falsae  positionis. 

Ar.  =  Theorematum  in  lihris  Archimedis  de  Sphaera  b°  cylindro 
declaratio. 

Ho.  =  Horologia  scioterica  in  piano,  geomelrice  delineandi  modus. 

The  abbreviated  titles  given  here  are,  of  course,  our  own.  The 
lists  of  tracts  added  to  the  Clavis  mathematicae  of  1631  in  its  later 


Principal  Works  19 

book  underwent  hardly  any  changes,  except  for  the  num- 
ber of  tracts  appended,  and  brief  explanatory  notes  added 
at  the  close  of  the  chapters  in  the  English  editions  of 
1694  and  1702.  The  1652  and  1667  editions  were  seen 
through  the  press  by  John  Wallis;  the  1698  impression 
contains  on  the  title-page  the  words:  Ex  Recognitione  D. 
Johannis  Wallis,  S.T.D.  Geometriae  Professoris  Saviliani. 

The  cost  of  publishing  may  be  a  matter  of  some  inter- 
est. When  arranging  for  the  printing  of  the  1667  edition 
of  the  Clavis,  Wallis  wrote  Collins:  "I  told  you  in  my  last 
what  price  she  [Mrs.  Lichfield]  expects  for  it,  as  I  have 
formerly  understood  from  her,  viz.,  £  40  for  the  impression, 
which  is  about  g^d.  a  book."^ 

As  compared  with  other  contemporary  works  on  alge- 
bra, Oughtred's  distinguishes  itself  for  the  amount  of 
symbolism  used,  particularly  in  the  treatment  of  geo- 
metric problems.  Extraordinary  emphasis  was  placed 
up  an  what  he  called  in  the  Clavis  the  "analytical  art."^ 

editions,  given  in  the  order  in  which  the  tracts  appear  in  each  edition, 
are  as  follows:  Clavis  of  1647,  Eq.,  An.,  Fa.,  Ho.;  Clavis  of  1648, 
Eq.,  An.,  Fa.,  Eu.,  So.;  Clavis  of  1652,  Eq.,  Eu.,  So.,  An.,  Fa., 
Ar.,  Ho.;  Clavis  of  1667,  Eq.,  Eu.,  So.,  An.,  Fa.,  Ar.,  Ho.;  Clavis  of 
1693  and  1698,  Eq.,  Eu.,  So.,  An.,  Fa.,  Ar.,  Ho.;  Clavis  of  1694  and 
1702,  Eq. 

The  title-page  of  the  Clavis  was  considerably  modified  after  the 
first  edition.  Thus,  the  1652  Latin  edition  has  this  title-page: 
Guilehni  Oiightred  Aetonensis ,  quondam  Collegii  Regalis  in  Cantabrigia 
Socii,  Clavis  mathematicae  denvo  limata,  sive  potius  fabricata.  Cum 
aliis  quihusdam  ejusdem  commentationihus ,  quae  in  sequenti  pagina 
recensentur.  Editio  tertia  auctior  b"  emendatior.  Oxoniae,  Excudehat 
Leon.     Lichfield,  Veneunt  apud  Tho.  Robinson.     1652. 

^  Rigaud,  op.  cit.,  Vol.  II,  p.  476. 

2  See,  for  instance,  the  Clavis  mathematicae  of  1652,  where  he 
expresses  himself  thus  (p.  4):  "Speciosa  haec  Arithmetica  arti 
Analyticae  (per  quam  ex  surnptione  quaesiti,  tanquam  noti, 
investigatur  quaesitum)  multo  accommodatior  est,  quam  ilia 
numerosa." 


20  William  Oughtred 

By  that  term  he  did  not  mean  our  modern  analysis  or 
analytical  geometry,  but  the  art  ''in  which  by  taking  the 
thing  sought  as  knowne,  we  finde  out  that  we  seeke."^ 
He  meant  to  express  by  it  condensed  processes  of  rigid, 
logical  deduction  expressed  by  appropriate  symbols,  as 
contrasted  with  mere  description  or  elucidation  by  pas- 
sages fraught  with  verbosity.  In  the  preface  to  the  first 
edition  (163 1)  he  says: 

In  this  little  book  I  make  known  ....  the  rules  relating 
to  fundamentals,  collected  together,  just  like  a  bundle,  and 
adapted  to  the  explanation  of  as  many  problems  as  possible. 

As  stated  in  this  preface,  one  of  his  reasons  for  publish- 
ing the  book,  is 

....  that  like  Ariadne  I  might  offer  a  thread  to  mathematical 
study  by  which  the  mysteries  of  this  science  might  be  revealed, 
and  direction  given  to  the  best  authors  of  antiquity,  EucUd, 
Archimedes,  the  great  geometrician  ApoUonius  of  Perga,  and 
others,  so  as  to  be  easily  and  thoroughly  understood,  their 
theorems  being  added,  not  only  because  to  many  they  are  the 
height  and  depth  of  mathematical  science  (I  ignore  the  would-be 
mathematicians  who  occupy  themselves  only  with  the  so-called 
practice,  which  is  in  reahty  mere  juggler's  tricks  with  instru- 
ments, the  surface  so  to  speak,  pursued  with  a  disregard  of  the 
great  art,  a  contemptible  picture),  but  also  to  show  with  what 
keenness  they  have  penetrated,  with  what  mass  of  equations, 
comparisons,  reductions,  conversions  and  disquisitions  these 
heroes  have  ornamented,  increased  and  invented  this  most 
beautiful  science. 

The  Clavis  opens  with  an  explanation  of  the  Hindu- 
Arabic  notation  and  of  decimal  fractions.  Noteworthy  is 
the  absence  of  the  words  "million,"  "bilhon,"  etc.  Though 
used  on  the  Continent  by  certain  mathematical  writers 
long  before  this,  these  words  did  not  become  current  in 

I  Oughtred,  The  Key  of  the  Mathematicks,  London,  1647,  p.  4. 


Principal  Works  21 

English  mathematical  books  until  the  eighteenth  century. 
The  author  was  a  great  admirer  of  decimal  fractions,  but 
failed  to  introduce  the  notation  which  in  later  centuries 
came  to  be  universally  adopted.  Oughtred  wrote  0.56 
in  this  manner  o|56;  the  point  he  used  to  designate  ratio. 
Thus  3:4  was  written  by  him  3 •4.  The  decimal  point 
(or  comma)  was  first  used  by  the  inventor  of  logarithms, 
John  Napier,  as  early  as  16 16  and  16 17.  Although 
Oughtred  had  mastered  the  theory  of  logarithms  soon  after 
their  publication  in  16 14  and  was  a  great  admirer  of 
Napier,  he  preferred  to  use  the  dot  for  the  designation  of 
ratio.  This  notation  of  ratio  is  used  in  all  his  mathe- 
matical books,  except  in  two  instances.  The  two  dots  (:) 
occur  as  symbols  of  ratio  in  some  parts  of  Oughtred 's 
posthumous  work,  Opuscula  mathematica  hactenus  inedita, 
Oxford,  1677,  but  may  have  been  due  to  the  editors  and 
not  to  Oughtred  himself.  Then  again  the  two  dots  (:) 
are  used  to  designate  ratio  on  the  last  two  pages  of  the 
tables  of  the  Latin  edition  of  Oughtred 's  Trigonometria 
of  1657.  In  all  other  parts  of  that  book  the  dot  (•)  is 
used.  Probably  someone  who  supervised  the  printing 
of  the  tables  introduced  the  (:)  on  the  last  two  pages, 
following  the  logarithmic  tables,  where  methods  of  inter- 
polation are  explained.  The  probability  of  this  conjecture 
is  the  stronger,  because-  in  the  English  edition  of  the 
Trigonometrie,  brought  out  the  same  year  (1657)  but  after 
the  Latin  edition,  the  notation  ( : )  at  the  end  of  the  book 
is  replaced  by  the  usual  (•)?  except  that  in  some  copies 
of  the  English  edition  the  explanations  at  the  end  are 
omitted  altogether. 

Oughtred  introduces  an  interesting,  and  at  the  same 
time  new,  feature  of  an  abbreviated  multiplication  and  an 
abbreviated  division  of  decimal  fractions.     On  this  point 


22  William  Oughtred 

he  took  a  position  far  in  advance  of  his  time.  The  part 
on  abbreviated  multiplication  was  rewritten  in  slightly 
enlarged  form  and  with  some  unimportant  alterations 
in  the  later  edition  of  the  Clavis.  We  give  it  as  it  occurs 
in  the  revision.  Four  cases  are  given.  In  finding  the 
product  of  246|9i4and35|27,  ''if  you  would 
have  the  Product  without  any  Parts"  2461914 
(without  any  decimal  part),  "set  the  place     72153 

of  Unity  of  the  lesser  under  the  place  of    

Unity  in  the  greater:    as  in  the  Example,"     '  ^       ' 
writing  the  figures  of  the  lesser  number  m  ^  ^ 

inverse  order.     From  the  example  it  will  be  ^  ^ 

•  I    7 

seen  that  he  begins  by  multiplying  by  3,  the     L 

right-hand  digit  of  the  multiplier.  In  the  8708 
first  edition  of  the  Clavis  he  began  with  7, 
the  left  digit.  Observe  also  that  he  ''carries"  the  nearest 
tens  in  the  product  of  each  lower  digit  and  the  upper  digit 
one  place  to  its  right.  For  instance,  he  takes  7X4=28 
and  carries  3 ,  then  he  finds  7X2+3=17  and  writes  down  1 7 . 
The  second  case  supposes  that  "you  would  have  the 
Product  with  some  places  of  parts"  (decimals),  say  4: 
"Set  the  place  of  Unity  of  the  lesser  Number  under  the 
Fourth  place  of  the  Parts  of  the  greater."  The  multi- 
plication of  246 1 9 14  by  35 1 27  is  now  performed  thus: 

2  4  6|9  I  4 


72  53 


74074200 

12345700 
493828 
172840 

87086568 


Principal  Works  23 

In  the  third  and  fourth  cases  are  considered  factors 
which  appear  as  integers,  but  are  in  reahty  decimals; 
for  instance,  the  sine  of  54°  is  given  in  the  tables  as  80902 
when  in  reality  it  is  .80902. 

Of  interest  as  regards  the  use  of  the  word  "parabola" 
is  the  following:  "The  Number  found  by  Division  is 
called  the  Quotient,  or  also  Parabola,  because  it  arises  out 
of  the  Application  of  a  plain  Number  to  a  given  Longitude, 
that  a  congruous  Latitude  may  be  found. "^  This  is  in 
harmony  with  etymological  dictionaries  which  speak  of  a 
parabola  as  the  application  of  a  given  area  to  a  given 
straight  line.  The  dividend  or  product  is  the  area;  the 
divisor  or  factor  is  the  line. 

Oughtred  gives  two  processes  of  long  division.  The 
first  is  identical  with  the  modern  process,  except  that  the 
divisor  is  written  below  every  remainder,  each  digit  of 
the  divisor  being  crossed  out  as  soon  as  it  has  been  used 
in  the  partial  multiplication.  The  second  method  of 
long  division  is  one  of  the  several  tjrpes  of  the  old  "scratch 
method."  This  antiquated  process  held  its  place  by  the 
side  of  the  modern  method  in  all  editions  of  the  Clavis. 
The  author  divides  467023  by  35710926425,  giving  the 
following  instructions:  "Take  as  many  of  the  first  Figures 
of  the  Divisor  as  are  necessary,  for  the  first  Divisor,  and 
then  in  every  following  particular  Division  drop  one  of 
the  Figures  of  the  Divisor  towards  the  Left  Hand,  till 
you  have  got  a  competent  Quotient. ' '  He  does  not  explain 
abbreviated  division  as  thoroughly  as  abbreviated  multi- 
plication. 

^  Clavis  1694,  p.  19,  and  the  Clavis  of  163 1,  p.  8. 


24  William  Oughtred 


17 


357 [0926425)   mm   (1307 |8o 
^^w^ 

WW 

m 

Oughtred  does  not  examine  the  degree  of  rehabiUty 
or  accuracy  of  his  processes  of  abbreviated  multipUcation 
and  division.  Here  as  in  other  places  he  gives  in  con- 
densed statement  the  mode  of  procedure,  without  further 
discussion. 

He  does  not  attempt  to  establish  the  rules  for  the  addi- 
tion, subtraction,  multiplication,  and  division  of  positive 
and  negative  numbers.  "If  the  Signs  are  both  alike,  the 
Product  will  be  afl5.rmative,  if  unlike,  negative";  then 
he  proceeds  to  applications.  This  attitude  is  superior  to 
that  of  many  writers  of  the  eighteenth  and  nineteenth 
centuries,  on  pedagogical  as  well  as  logical  grounds: 
pedagogically,  because  the  beginner  in  the  study  of  algebra 
is  not  in  a  position  to  appreciate  an  abstract  train  of 
thought,  as  every  teacher  well  knows,  and  derives  better 
intellectual  exercise  from  the  applications  of  the  rules  to 
problems;  logically,  because  the  rule  of  signs  in  multi- 
plication does  not  admit  of  rigorous  proof,  unless  some 
other  assumption  is  first  made  which  is  no  less  arbitrary 
than  the  rule  itself.  It  is  well  known  that  the  proofs 
of  the  rule  of  signs  given  by  eighteenth-century  writers 
are  invalid.  Somewhere  they  involve  some  surreptitious 
assumption.  This  criticism  applies  even  to  the  proof 
given  by  Laplace,  which  tacitly  assumes  the  distributive 
law  in  multiplication. 


Principal  Works  25 

A  word  should  be  said  on  Oughtred's  definition  of  + 
and  — .  He  recognizes  their  double  function  in  algebra  by 
saying  (Clavis,  163 1,  p.  2):  ''Signum  additionis,  sive 
affirmationis,  est  +  plus"  and  "Signum  subductionis, 
sive  negationis  est  —  minus."  They  are  symbols  which 
indicate  the  quality  of  numbers  in  some  instances  and 
operations  of  addition  or  subtraction  in  other  instances. 
In  the  1694  edition  of  the  Clavis,  thirty-four  years  after 
the  death  of  Oughtred,  these  symbols  are  defined  as  signi- 
fying operations  only,  but  are  actually  used  to  signify  the 
quality  of  numbers  as  well.  In  this  respect  the  1694 
edition  marks  a  recrudescence. 

The  characteristic  in  the  Clavis  that  is  most  striking 
to  a  modern  reader  is  the  total  absence  of  indexes  or  expo- 
nents. There  is  much  discussion  in  the  leading  treatises 
of  the  latter  part  of  the  sixteenth  and  the  early  part  of  the 
seventeenth  century  on  the  theory  of  indexes,  but 
the  modern  exponential  notation,  a^,  is  of  later  date. 
The  modern  notation,  for  positive  integral  exponents,  first 
appears  in  Descartes'  Geometrie,  1637;  fractional  and 
negative  exponents  were  first  used  in  the  modern  form 
by  Sir  Isaac  Newton,  in  his  announcement  of  the  binomial 
formula,  in  a  letter  written  in  1676.  This  total  absence 
of  our  modern  exponential  notation  in  Oughtred's  Clavis 
gives  it  a  strange  aspect.  Like  Vieta,  Oughtred  uses  ordi- 
narily the  capital  letters,  ^,  ^,  C,  .  .  .  .  to  designate 
given  numbers;  A^  is  written  Aq,  A^  is  written  Ac;  for 
^"f,  A^,  A^  he  has,  respectively,  Aqq,  Aqc,  Ace.  Only  on 
rare  occasions,  usually  when  some  parallelism  in  notation 
is  aimed  at,  does  he  use  small  letters^  to  represent  numbers 
or    magnitudes.     Powers    of   binomials    or   polynomials 

^  See  for  instance,  Oughtred's  Elementi  decivii  Euclidis  dedaratio, 
1652,  p.  I,  where  he  uses  A  and  E,  and  also  a  and  e. 


26  William  Oughtred 

are  marked  by  prefixing  the  capital  letters  Q  (for  square), 
C  (for  cube),  QQ  (for  the  fourth  power),  QC  (for  the  fifth 
power),  etc. 

Oughtred  does  not  express  aggregation  by  ( ) .  Par- 
entheses had  been  used  by  Girard,  and  by  Clavius  as 
early  as  1609,^  but  did  not  come  into  general  use  in 
mathematical  language  until  the  time  of  Leibniz  and  the 
Bernoullis.  Oughtred  indicates  aggregation  by  writing 
a  colon  (:)  at  both  ends.  Thus,  (2:^  — £: means  with 
him  {A  —  Ey.  Similarly,  ^  q'.A-\-E\  means  \^\A-\-E). 
The  two  dots  at  the  end  are  frequently  omitted  when  the 
part  affected  includes  all  the  terms  of  the  polynomial  to 
the  end.  Thus,  C\A^B-E=.  .  means  {aVB-Ey=.  . 
There  are  still  further  departures  from  this  notation,  but 
they  occur  so  seldom  that  we  incline  to  the  interpretation 
that  they  are  simply  printer's  errors.  For  proportion 
Oughtred  uses  the  symbol  ( : : ) .  The  proportion  a:b  = 
c:d  appears  in  his  notation  a-hwc-d.  Apparently,  a 
proportion  was  not  fully  recognized  in  this  day  as  being 
the  expression  of  an  equality  of  ratios.  That  probably 
explains  why  he  did  not  use  =  here  as  in  the  notation  of 
ordinary  equations.  Yet  Oughtred  must  have  been  very 
close  to  the  interpretation  of  a  proportion  as  an  equahty; 
for  he  says  in  his  Elementi  decimi  Euclidis  declaratio, 
"proportio,  sive  ratio  aequalis  : :  "  That  he  introduced 
this  extra  symbol  when  the  one  for  equality  was  sufficient 
is  a  misfortune.  Simplicity  demands  that  no  unnecessary 
symbols  be  introduced.  However,  Oughtred 's  symbolism 
is  certainly  superior  to  those  which  preceded.  Consider 
the  notation  of  Clavius.^^    He  wrote  2o:6o  =  4:x,  .t=i2, 

^  See  Christophor  i  Clavii  Bamhergensis  Operum  maihematicorum, 
tomus  secuiidus,  Moguntiae,  M.DC.XI,  algebra,  p.  39. 

^  Christophori  Clavii  operum  mathematicorum  Tomus  Secmidus, 
Moguntiae,  M.DC.XI,  Epitome  arithmeticae,  p.  36. 


Principal  Works  27 

thus:  "2o-6o«4?  jiunt  12."  The  insufficiency  of  such  a 
notation  in  the  more  involved  expressions  frequently 
arising  in  algebra  is  readily  seen.  Hence  Oughtred's 
notation  ( : : )  was  early  adopted  by  English  mathema- 
ticians. It  was  used  by  John  Wallis  at  Oxford,  by  Samuel 
Foster  at  Gresham  College,  by  James  Gregory  of  Edin- 
burgh, by  the  translators  into  English  of  Rahn's  algebra, 
and  by  many  other  early  writers.  Oughtred  has  been 
credited  generally  with  the  introduction  of  St.  Andrew's 
cross  X  as  the  symbol  for  multiplication  in  the  Clavis  of 
163 1.  We  have  discovered  that  this  symbol,  or  rather 
the  letter  x  which  closely  resembles  it,  occurs  as  the  sign 
of  multiplication  thirteen  years  earlier  in  an  anonymous 
''Appendix  to  the  Logarithmes,  shewing  the  practise  of 
the  Calculation  of  Triangles  etc."  to  Edward  Wright's 
translation  of  John  Napier's  Description  published  in  1618.^ 
Later  we  shall  give  our  reasons  for  believing  that  Oughtred 
is  the  author  of  that  "Appendix."  The  X  has  survived 
as  a  symbol  of  multiphcation. 

Another  symbol  introduced  by  Oughtred  and  found  in 
modern  books  is  ^,  expressing  difference;  thus  C^D 
signifies  the  difference  between  C  and  D,  even  when  D  is 
the  larger  number.^  This  symbol  was  used  by  John 
Wallis  in  1657.2 

Oughtred  represented  in  symbols  also  certain  com- 
posite expressions,  as  for  instance  A-\-E  =  Z,  A—E=X, 
where  A  is  greater  than  E.  He  represented  by  a  sym- 
bol also  each  of  the  following:  A'+E',  A^+E^,  A'-E', 
A^-EK 

^  See  F.  Cajori,  "The  Cross  X  as  a  Symbol  of  Muhiplication," 
in  Nature,  Vol.  XCIV  (1914),  p.  363. 

^  See  Elementi  decimi  Euclidis  declaratio,  1652,  p.  2. 

3  See  Johannis  Wallisii  Operum  mathematicorum  pars  prima, 
Oxonii,  1657,  p.  247. 


28  William  Oughtred 

Oughtred  practically  translated  the  tenth  book  of 
Euclid  from  its  ponderous  rhetorical  form  into  that  of 
brief  symbolism.  An  appeal  to  the  eye  was  a  passion  with 
Oughtred.  The  present  writer  has  collected  the  different 
mathematical  symbols  used  by  Oughtred  and  has  found 
more  than  one  hundred  and  fifty  of  them. 

The  differences  between  the  seven  different  editions  of 
the  Clavis  lie  mainly  in  the  special  parts  appended  to  some 
editions  and  dropped  in  the  latest  editions.  The  part 
which  originally  constituted  the  Clavis  was  not  materially 
altered,  except  in  two  or  three  of  the  original  twenty 
chapters.  These  changes  were  made  in  the  editions  of 
1647  and  1648.  After  the  first  edition,  great  stress  was 
laid  upon  the  theory  of  indices  upon  the  very  first  page, 
as  also  in  passages  farther  on.  Of  course,  Oughtred  did 
not  have  our  modern  notation  of  indices  or  exponents, 
but  their  theory  had  been  a  part  of  algebra  and  arithmetic 
for  some  time.  Oughtred  incorporated  this  theory  in  his 
brief  exposition  of  the  Hindu-Arabic  notation  and  in  his 
explanation  of  logarithms.  As  previously  pointed  out, 
the  last  three  chapters  of  the  163 1  edition  were  consider- 
ably rearranged  in  the  later  editions  and  combined  into 
two  chapters,  so  that  the  Clavis  proper  had  nineteen 
chapters  instead  of  twenty  in  the  additions  after  the  first. 
These  chapters  consisted  of  applications  of  algebra  to 
geometry  and  were  so  framed  as  to  constitute  a  severe 
test  of  the  student's  grip  of  the  subject.  The  very  last 
problem,  deals  with  the  division  of  angles  into  equal  parts. 
He  derives  the  cubic  equation  upon  which  the  trisection 
depends  algebraically,  also  the  equations  of  the  fifth  degree 
and  seventh  degree  upon  which  the  divisions  of  the  angle 
into  5  and  7  equal  parts  depend,  respectively.  The 
exposition  was  severely  brief,  yet  accurate.     He  did  not 


Principal  Works  29 

believe  in  conducting  the  reader  along  level  paths  or  along 
slight  inclines.  He  was  a  guide  for  mountain-climbers, 
and  woe  unto  him  who  lacked  nerve. 

Oughtred  lays  great  stress  upon  expansions  of  powers  of 
a  binomial.  He  makes  use  of  these  expansions  in  the 
solution  of  numerical  equations.  To  one  who  does  not 
specialize  in  the  history  of  mathematics  such  expansions 
may  create  surprise,  for  did  not  Newton  invent  the 
binomial  theorem  after  the  death  of  Oughtred?  As  a 
matter  of  fact,  the  expansions  of  positive  integral  powers 
of  a  binomial  were  known  long  before  Newton,  not  only 
to  seventeenth-century  but  even  to  eleventh-century 
mathematicians.  Oughtred's  Clavis  of  163 1  gave  the 
binomial  coefficients  for  all  powers  up  to  and  including 
the  tenth.  What  Newton  really  accomplished  was  the 
generalization  of  the  binomial  expansion  which  makes  it 
applicable  to  negative  and  fractional  exponents  and  con- 
verts it  into  an  infinite  series. 

As  a  specimen  of  Oughtred's  style  of  writing  we  quote 
his  solution  of  quadratic  equations,  accompanied  by  a 
translation  into  English  and  into  modern  mathematical 
symbols. 

As  a  preliminary  step^  he  lets 

Z  =  ^+£and^>E; 

he  lets  also  X  =  A—E.  From  these  relations  he  obtains 
identities  which,  in  modern  notation,  are  \Z^—AE  = 
{iZ-Ey  =  iX\  Now,  if  we  know  Z  and  AE,  we 
can  find  JX.  Then  i{Z-{-X)=A,  and  i(Z-X)=E, 
and 

A=iZ+VlZ'-AE. 
^Clavis  of  1631,  chap,  xix,  sec.  5,  p.  50. 


30  William  Oughtred 

Having  established  these  preliminaries,  he  proceeds 
thus: 

Datis  igitur  hnea  inaequahter  secta  Z  (lo),  &  rectangulo 
sub  segmentis  AE  (21)  qui  gnomon  est:  datur  semidifferentia 
segmentorum  ^X:  &  per  consequens  ipsa  segmenta.  Nam 
ponatur  alterutrum  segmentum  A :  alterum  erit  Z—A:  Rec- 
tangulum  auctem  est  ZA—Aq  =  AE.  Et  quia  dantur  Z  & 
AE:  estque  \Zq-AE=lXq:  &  per  5c.  18,  |Z+§X=^:  & 
fZ— ^X=£:  Aequatio  sic  resoluetur:  \Z^V q:\Zq— AE: 
_  .  fmaius  segment 
\minus  segment. 

Itaque  proposita  equatione,  in  qua  sunt  tres  species  aequa- 
liter  in  ordine  tabellae  adscendentes,  altissima  autem  species 
ponitur  negata:  Magnitudo  data  coefficiens  mediam  speciem 
est  linea  bisecanda:  &  magnitudo  absoluta  data,  ad  quam  sit 
aequatio,  est  rectangulum  sub  segmentis  inaequalibus,  sine 
gnomon:  vt  ZA—Aq  =  AE:  in  numeris  autem  10/— /g  =  2i: 
Estque  A,  vel  il,  alterutrum  segmentum  inaequale.  Inuenitur 
autem  sic: 

Z  . 

Dimidiata   coefficiens  median   speciem  est   —   (5);    cuius 

quadratum  est  —   (25):     ex  hoc  toUe  AE   (21)  absolutum: 
4 

eritque  —  —AE  (4)  quadratum  semidifferentiae  segmentorum: 
4 

latus    huius  quadratum  \^' q\  ——AE  (2)    est    semidifferentia: 

Z 

quam     si     addas     ad     —  (5)     semissem     coefficientis,     sive 

lineae  bisecandae,   erit  maius  segment.;     sin  detrahas,   erit 

-^ .      Z       ,    Zn      ,  ^        A  maius  segmentuni 

mmus segment :  D ICO -±1/(7: AE-.^AK 

°  2         ^    \  [nunus  segmentum. 

We  translate  the  Latin  passage,  nsing  the  modern 
exponential  notation  and  parentheses,  as  follows: 

Given  therefore  an  unequally  divided  line  Z  (10),  and  a 
rectangle  beneath  the  segments  AE  (21)  which  is  a  gnomon. 


Principal  Works  31 

Half  the  difference  of  the  segments  \X  is  given,  and  conse- 
quently the  segment  itself.  For,  if  one  of  the  two  segments  is 
placed  equal  to  A,  the  other  will  be  Z—A.  Moreover,  the 
rectangle  is  ZA  —A^  =  AE.  And  because  Z  and  AE  are  given, 
and  there  is  \Z^-AE  =  \X^,  and  by  5c.  18,  \Z-\-lX  =  A,  and 
^Z—\X=E,  the  equation  will  be  solved  thus:   ^Z^V{\Z^— 

.  ■r.x       A  niajor  segment 
AE)=A{     . 

^         [mmor  segment. 

And  so  an  equation  having  been  proposed  in  which  three 
species  (terms)  are  in  equally  ascending  powers,  the  highest 
species,  moreover,  being  negative,  the  given  magnitude  which 
constitutes  the  middle  species  is  the  line  to  be  bisected.  And 
the  given  absolute  magnitude  to  which  it  is  equal  is  the  rec- 
tangle beneath  the  unequal  segments,  without  gnomon.  As 
ZA—A^  =  AE,  or  in  numbers,  io:k— :\;^  =  2i.  And  ^  or  x  is 
one  of  the  two  unequal  segments.     It  may  be  found  thus: 

Z  Z^ 

The  half  of  the  middle  species  is  —  (5),  its  square  is  —  (25). 

2  4 

Z^ 

From  it  subtract  the  absolute  term  AE  (21),  and  ——^£(4) 

4 

will  be  the  square  of  half  the  difference  of  the  segments. 
The  square  root  of  this,  V    ( — J  —AE\  (2),  is  half  the  differ- 

ence.     If  you  add  it  to  half  the  coefi&cient  —  (5),  or  half  the  line 

to  be  bisected,  the  longer  segment  is  obtained;  if  you  subtract 
it,  the  smaller  segment  is  obtained.    I  say : 


2         V  4  /         1^  minor  segment. 


The  quadratic  equation  Aq-\-ZA  =AE  receives  similar 
treatment.  This  and  the  preceding  equation,  ZA—Aq  = 
AE,  constitute  together  a  solution  of  the  general  quadratic 
equation,  x'^-{-ax  =  b,  provided  that  E  or  Z  are  not  re- 
stricted to  positive  values,   but  admit  of  being  either 


32  William  Oughtred 

positive  or  negative,  a  case  not  adequately  treated  by 
Oughtred.  Imaginary  numbers  and  imaginary  roots  re- 
ceive no  consideration  whatever. 

A  notation  suggested  by  Vieta  and  favored  by  Girard 
made  vowels  stand  for  unknowns  and  consonants  for 
knowns.  This  conventionality  was  adopted  by  Oughtred 
in  parts  of  his  algebra,  but  not  throughout.  Near  the 
beginning  he  used  Q  to  designate  the  unknown,  though 
usually  this  letter  stood  with  him  for  the  ''square"  of 
the  expression  after  it.^ 

It  is  of  some  interest  that  Oughtred  used  ^  to  signify 

the  ratio  of  the  circumference  to  the  diameter  of  a  circle. 
Very  probably  this  notation  is  the  forerunner  of  the  7r  = 
3.14159  ....  used  in  1706  by  William  Jones.     Ought- 

red  first  used  -s  in  the  1647  edition  of  the  Clavis  mathe- 

o 

maticae.  In  the  1652  edition  he  says,  ''Si  in  circulo  sit 
7.22:  :8'7r:  1113. 355:erit  8-7r::2  R.P:  periph."  This 
notation  was  adopted  by  Isaac  Barrow,  who  used  it  exten- 

sively.     David  Gregory^  used  -  in  1697,  and  De  Moivre^ 

r 

C 

used  -  about  1697,  to  designate  the  ratio  of  the  circumfer- 
r 

ence  to  the  radius. 

^  We  have  noticed  the  representation  of  known  quantities  by 
consonants  and  the  unknown  by  vowels  in  Wingate's  Arithmetick 
made  easie,  edited  by  John  Kersey,  London,  1650,  algebra,  p.  382; 
and  in  the  second  part,  section  19,  of  Jonas  ]\Ioore's  Arithmetick  in 
two  parts,  London,  1660,  Moore  suggests  as  an  alternative  the  use 
of  s,  y,  X,  etc.,  for  the  unknowns.  The  practice  of  representing 
unknowns  by  vowels  did  not  spread  widely  in  England. 

""  Philosophical  Transactions,  Vol.  XIX,  No.  231,  London,  p.  652. 

3  Ihid.,  Vol.  XIX,  p.  56. 


Principal  Works  33 

We  quote  the  description  of  the  Clavis  that  was  given 
by  Oughtred's  greatest  pupil,  John  WaUis.  It  contains 
additional  information  of  interest  to  us.  Wallis  devotes 
chap.  XV  of  his  Treatise  of  Algebra,  London,  1685,  pp.  67- 
69,  to  Mr.  Oughtred  and  his  Clavis,  saying: 

Mr.  William  Oughtred  (our  Country-man)  in  his  Clavis 
Mathematicae,  (or  Key  of  Mathematicks,)  first  published  in  the 
Year  1631,  follows  Vieta  (as  he  did  Diophantus)  in  the  use  of 
the  Cossick  Denominations;  omitting  (as  he  had  done)  the 
names  of  Sursolids,  and  contenting  himself  with  those  of  Square 
and  Cube,  and  the  Compounds  of  these. 

But  he  doth  abridge  Vieta's  Characters  or  Species,  using 
only  the  letters  q,  c,  &c.  which  in  Vieta  are  expressed  (at  length) 
by  Quadrate,  Cube,  &c.  For  though  when  Vieta  first  intro- 
duced this  way  of  Specious  Arithmetick,  it  was  more  necessary 
(the  thing  being  new,)  to  express  it  in  words  at  length:  Yet 
when  the  thing  was  once  received  in  practise,  Mr.  Oughtred 
(who  affected  brevity,  and  to  deliver  what  he  taught  as  briefly 
as  might  be,  and  reduce  all  to  a  short  view,)  contented  himself 
with  single  Letters  instead  of  those  words. 

Thus  what  Vieta  would  have  written 

A  Quadrate,  into  B  Cube,  _      ,      t.^  t^7 
CDESoHd, ^^^^^  ^'  ^^  ^^^^'' 

would  with  him  be  thus  expressed 

CDE    ^^' 

And  the  better  to  distinguish  upon  the  first  view,  what 
quantities  were  Known,  and  what  Unknown,  he  doth  (usually) 
denote  the  Known  to  Consonants,  and  the  Unknown  by  Vowels; 
as  Vieta  (for  the  same  reason)  had  done  before  him. 

He  doth  also  (to  very  great  advantage)  make  use  of  several 
Ligatures,  or  Compendious  Notes,  to  signify  Summs,  Differ- 
ences, and  Rectangles  of  several  Quantities.    As  for  instance, 


34  William  Oughtred 

Of  two  Quantities  A  (the  Greater),  and  E  (the  Lesser),  the  Sum 
he  calls  Z,  the  Difference  X,  the  Rectangle  AE 

Which  being  of  (almost)  a  constant  signification  with  him 
throughout,  do  save  a  great  circumlocution  of  words,  (each 
Letter  serving  instead  of  a  Definition;)  and  are  also  made  use 
of  (with  very  great  advantage)  to  discover  the  true  nature  of 
divers  intricate  Operations,  arising  from  the  various  composi- 
tions of  such  Parts,  Sums,  Differences,  and  Rectangles;  (of 
which  there  is  great  plenty  in  his  Clavis,  Cap.  ii,  i6,  i8,  19. 
and  elsewhere,)  which  without  such  Ligatures,  or  Compendious 
Notes,  would  not  be  easily  discovered  or  apprehended 

I  know  there  are  who  find  fault  with  his  Clavis,  as  too  ob- 
scure, because  so  short,  but  without  cause;  for  his  words  be 
always  full,  but  not  Redundant,  and  need  only  a  Httle  atten- 
tion in  the  Reader  to  weigh  the  force  of  every  word,  and  the 
Syntax  of  it;  ...  .  And  this,  when  once  apprehended,  is 
much  more  easily  retained,  than  if  it  were  expressed  with  the  pro- 
lixity of  some  other  Writers;  where  a  Reader  must  first  be  at 
the  pains  to  weed  out  a  great  deal  of  superfluous  Language, 
that  he  may  have  a  short  prospect  of  what  is  material;  which 
is  here  contracted  for  him  in  a  short  Synopsis 

Mr.  Oughtred  in  his  Clavis,  contents  himself  (for  the  most 
part)  with  the  solution  of  Quadratick  Equations,  without  pro- 
ceeding (or  very  sparingly)  to  Cubick  Equations,  and  those  of 
Higher  Powers;  having  designed  that  Work  for  an  Introduction 
into  Algebra  so  far,  leaving  the  Discussion  of  Superior  Equa- 
tions for  another  work He  contents  himself  likewise  in 

Resolving  Equations,  to  take  notice  of  the  Affirmative  or  Posi- 
tive Roots;  omitting  the  Negative  or  Ablative  Roots,  and  such  as 
are  called  Imaginary  or  Impossible  Roots.  And  of  those  which, 
he  calls  Ambiguous  Equations,  (as  having  more  Affirmative 
Roots  than  one,)  he  doth  not  (that  I  remember)  any  where  take 
notice  of  more  than  Two  Af&rmative  Roots:  (Because  in 
Quadratick  Equations,  which  are  those  he  handleth,  there  are 
indeed  no  more.)  Whereas  yet  in  Cubick  Equations,  there  may 
be  Three,  and  in  those  of  Higher  Powers,  yet  more.    Which 


Principal  Works  35 

Vieta  was  well  aware  of,  and  mentioneth  in  some  of  his  Writings; 
and  of  which  Mr.  Oughtred  could  not  be  ignorant. 

"circles  of  proportion"  and  ''trigonometrie" 

Oughtred  wrote  and  had  published  three  important 
mathematical  books,  the  Clavis,  the  Circles  of  Proportion,^ 
and  a  Trigonometrie.^  This  last  appeared  in  the  year  1657 
at  London,  in  both  Latin  and  English. 

It  is  claimed  that  the  trigonometry  was  '^neither 
finished  nor  published  by  himself,  but  collected  out  of 
his  scattered  papers;  and  though  he  connived  at  the 
printing  it,  yet  imperfectly  done,  as  appears  by  his  MSS.; 
and  one  of   the  printed  Books,  corrected  by  his  own 

^  There  are  two  titie-pages  to  the  edition  of  1632.  The  first  title- 
page  is  as  follows:  The  Circles  oj  Proportion  and  The  Horizotitall 
Instrument.  Both  invented,  and  the  vses  of  both  Written  i?i  Latine  by 
Mr.  W.  0.  Translated  into  English:  and  set  forth  for  the  publique 
benefit  by  William  Forster.  London.  Printed  for  Elias  Allen  maker 
of  these  and  all  other  mathematical  Instruments,  and  are  to  be  sold  at 
his  shop  over  against  St.  Clements  church  with  out  Temple-barr.  i6j2. 
T.  Cecill  Sculp. 

In  1633  there  was  added  the  following,  with  a  separate  title-page: 
An  addition  vnto  the  Vse  of  the  Instrvment  called  the  Circles  of  Propor- 
tion  London,  1633,  this  being  followed  by  Oughtred's  To  the 

English  Gentrie  etc.  In  the  British  Museum  there  is  a  copy  of  another 
impression  of  tht  Circles  of' Proportion,  dated  i62,g,vn\h\hQ.  Addition 
vnto  the  Vse  of  the  Instrument  etc.,  bearing  the  original  date,  1633, 
and  with  the  epistle,  To  the  English  Gentrie,  etc.,  inserted  imme- 
diately after  Forster's  dedication,  instead  of  at  the  end  of  the  volume. 

2  The  complete  title  of  the  English  edition  is  as  follows :  Trigono- 
metric, or,  The  manner  of  calculating  the  Sides  and  Angles  of  Triangles, 
by  the  Mathematical  Canon,  demonstrated.  By  William  Oughtred 
Etonens.  And  published  by  Richard  Stokes  Fellow  of  Kings  Colled ge  in 
Cambridge,  and  Arthur  Haughton  Gentleman.  London,  Printed  by 
R.  and  W.  Leybourn,  for  Thomas  Johnson  at  the  Coldest  Key  in  St. 
Pauls  Church-yard.    M.DC.LVII. 


36  William  Oughtred 

Hand."^    Doubtless  more  accurate  on  this  point  is  a  letter 
of  Richard  Stokes  who  saw  the  book  through  the  press: 

1  have  procured  your  Trigonometry  to  be  written  over  in  a 
fair  hand,  which  when  finished  I  will  send  to  you,  to  know  if  it 
be  according  to  your  mind;  for  I  intend  (since  you  were  pleased 
to  give  your  assent)  to  endeavour  to  print  it  with  Mr.  Briggs 
his  Tables,  and  so  soon  as  I  can  get  the  Prutenic  Tables  I  will 
turn  those  of  the  sun  and  moon,  and  send  them  to  you.^ 

In  the  preface  to  the  Latin  edition  Stokes  writes: 

Since  this  trigonometry  was  written  for  private  use  without 
the  intention  of  having  it  published,  it  pleased  the  Reverend 
Author,  before  allowing  it  to  go  to  press,  to  expunge  some  things, 
to  change  other  things  and  even  to  make  some  additions  and 
insert  more  lucid  methods  of  exposition. 

This  much  is  certain,  the  Trigonometry  bears  the  im- 
press characteristic  of  Oughtred.  Like  all  his  mathemati- 
cal writings,  the  book  was  very  condensed.  Aside  from 
the  tables,  the  text  covered  only  36  pages.  Plane  and 
spherical  triangles  were  taken  up  together.  The  treatise 
is  known  in  the  history  of  trigonometry  as  among  the 
very  earliest  works  to  adopt  a  condensed  symbolism  so 
that  equations  involving  trigonometric  functions  could 
be  easily  taken  in  by  the  eye.  In  the  work  of  1657,  con- 
tractions are  given  as  follows:  5  =  sine,  /  =  tangent,  ^e  = 
secant,  5  C(?  =  cosine  (sine  complement),  /  f;o  =  cotangent, 
56  CO  =  cosecant,  /(?^= logarithm,  Z  cru  =  sum.  of  the  sides 
of  a  rectangle  or  right  angle,  X  crw  =  difference  of  these 
sides.  It  has  been  generally  overlooked  by  historians 
that  Oughtred  used  the  abbreviations  of  trigonometric 
functions,  named  above,  a  quarter  of  a  century  earlier, 

^  Jer.  Collier,  The  Great  Historical,  Geographical,  Genealogical  and 
Poetical  Dictionary,  Vol.  II,  London,  1701,  art.  "Oughtred." 

2  Rigaud  op.  cit.,  Vol.  I,  p.  82. 


Principal  Works  37 

in  his  Circles  of  Proportion,  1632,  1633.  Moreover,  he 
used  sometimes  also  the  abbreviations  which  are  current 
at  the  present  time,  namely  sin  =  sine,  tan  =  tangent,  sec  = 
secant.  We  know  that  the  Circles  of  Proportion  existed 
in  manuscript  many  years  before  they  were  published. 
The  symbol  sv  for  sinus  versus  occurs  in  the  Clavis  of  1631. 
The  great  importance  of  well-chosen  symbols  needs  no 
emphasis  to  readers  of  the  present  day.  With  reference 
to  Oughtred's  trigonometric  symbols.  Augustus  De 
Morgan  said: 

This  is  so  very  important  a  step,  simple  as  it  is,  that  Euler 
is  justly  held  to  have  greatly  advanced  trigonometry  by  its 
introduction.  Nobody  that  we  know  of  has  noticed  that 
Oughtred  was  master  of  the  improvement,  and  willing  to  have 
taught  it,  if  people  would  have  learnt.^ 

We  find,  however,  that  even  Oughtred  cannot  be  given 
the  whole  credit  in  this  matter.  By  or  before  1631 
several  other  writers  used  abbreviations  of  the  trigo- 
nometric functions.  As  early  as  1624  the  contractions 
sin  for  sine  and  tan  for  tangent  appear  on  the  drawing 
representing  Gunter's  scale,  but  Gunter  did  not  use  them 
in  his  books,  except  in  the  drawing  of  his  scale.^  A  closer 
competitor  for  the  honor  of  first  using  these  trigonometric 
abbreviations  is  Richard  Norwood  in  his  Trigonometric, 
London,  1631,  where  s  stands  for  sine,  t  for  tangent,  sc 
for  sine  complement  (cosine),  tc  for  tangent  complement 
(cotangent),  and  sec  for  secant.  Norwood  was  a  teacher 
of  mathematics  in  London  and  a  well-known  writer  of 
books  on  navigation.     Aside  from  the  abbreviations  just 

^  A.  De  Morgan,'  Budget  of  Paradoxes,  London,  1872,  p.  451;  2d 
ed.,  Chicago,  1915,  Vol.  II,  p.  303. 

2  E.  Gunter,  Description  and  Use  of  the  Sector,  the  Crosse-stafe  and 
other  Instruments,  London,  1624,  second  book,  p.  31. 


38  William  Oughtred 

cited  Norwood  did  not  use  nearly  as  much  symbolism 
in  his  mathematics  as  did  Oughtred. 

Mention  should  be  made  of  trigonometric  symbols 
used  even  earlier  than  any  of  the  preceding,  in  "An 
Appendix  to  the  Logarithmes,  shewing  the  practise  of  the 
Calculation  of  Triangles,  etc.,"  printed  in  Edward  Wright's 
edition  of  Napier's  A  Description  of  the  Admirable  Table 
of  Logarithmes,  London,  1618.  We  referred  to  this  "Ap- 
pendix" in  tracing  the  origin  of  the  sign  X.  It  contains, 
on  p.  4,  the  following  passage :  "  For  the  Logarithme  of  an 
arch  or  an  angle  I  set  before  (s),  for  the  antilogarithme  or 
compliment  thereof  (5*)  and  for  the  Differential  (0."  In 
further  explanation  of  this  rather  unsatisfactory  passage, 
the  author  (Oughtred  ?)  says,  "As  for  example:  sB-\-BC  = 
CA.  that  is,  the  Logarithme  of  an  angle  B.  at  the  Base 
of  a  plane  right-angled  triangle,  increased  by  the  addition 
of  the  Logarithm  of  BC,  the  hypothenuse  thereof,  is  equall 
to  the  Logarithme  of  CA  the  cathetus." 

Here  "logarithme  of  an  angle  ^"  evidently  means 
"log  sin  j5,"  just  as  with  Napier,  "Logarithms  of  the 
arcs"  signifies  really  "Logarithms  of  the  sines  of  the 
angles."  In  Napier's  table,  the  numbers  in  the  column 
marked  "Differentiae"  signify  log  sine  minus  log  cosine 
of  an  angle;  that  is,  the  logarithms  of  the  tangents.  This 
explains  the  contraction  (t)  in  the  "Appendix."  The 
conclusion  of  all  this  is  that  as  early  as  16 18  the  signs  s,  s*, 
were  used  for  sine,  cosine,  and  tangent,  respectively. 

John  Speidell,  in  his  Breefe  Treatise  of  Sphaericall 
Triangles,  London,  1627,  uses  Si.  for  sine,  T.  and  Tan 
for  tangent,  Se.  for  secant,  Si.Co.  for  cosine,  Se.  Co.  for 
cosecant,  T.  Co.  for  cotangent. 

The  innovation  of  designating  the  sides  and  angles  of 
a  triangle  by  ^,  ^,  C,  and  a,  b,  c,  so  that  A  was  opposite 


Principal  Works  39 

a,  B  opposite  h,  and  C  opposite  c,  is  attributed  to  Leonard 
Euler  (1753),  but  was  first  used  by  Richard  Rawlinson 
of  Queen's  College,  Oxford,  sometimes  after  1655  and 
before  1668.  Oughtred  did  not  use  Rawlinson's  notation.^ 
In  trigonometry  English  writers  of  the  first  half  of  the 
seventeenth  century  used  contractions  more  freely  than 
their  continental  contemporaries;  even  more  freely,  indeed, 
than  English  writers  of  a  later  period.  Von  Braunmiihl, 
the  great  historian  of  trigonometry,  gives  Oughtred  much 
praise  for  his  trigonometry,  and  points  out  that  half  a 
century  later  the  army  of  writers  on  trigonometry  had 
hardly  yet  reached  the  standard  set  by  Oughtred's 
analysis.^  Oughtred  must  be  credited  also  with  the  first 
complete  proof  that  was  given  to  the  first  two  of  ''Napier's 
analogies."  His  trigonometry  contains  seven-place  tables 
of  sines,  tangents,  and  secants,  and  six-place  tables  of 
logarithmic  sines  and  tangents ;  also  seven-place  logarith- 
mic tables  of  numbers.  At  the  time  of  Oughtred  there 
was  some  agitation  in  favor  of  a  wider  introduction  of 
decimal  systems.  This  movement  is  reflected  in  those 
tables  which  contain  the  centesimal  division  of  the  degree, 
a  practice  which  is  urged  for  general  adoption  in  our  own 
day,  particularly  by  the  French. 

SOLUTION   OF   NUMERICAL  EQUATIONS 

In  the  solution  of  numerical  equations  Oughtred  does 
not  mention  the  sources  from  which  he  drew,  but  the 
method  is  substantially  that  of  the  great  French  alge- 
braist Vieta,  as  explained  in  a  publication  which  appeared 

^  F.  Cajori,  "On  the  History  of  a  Notation  in  Trigonometry," 
Nature,  Vol.  XCIV,  1915,  pp.  642,  643. 

2  A.  von  Braunmuhl,  Geschichte  der  Trigonometrie,  2.  Teil,  Leipzig, 
1903,  pp.  42,  91. 


40  William  Oughtred 

in  1600  in  Paris  under  the  title,  De  numerosa  potestatum 
piirarum  atque  adfedarum  ad  exegesin  resolutione  tractatus. 
In  view  of  the  fact  that  Vieta's  process  has  been  described 
inaccurately  by  leading  modern  historians  including  H. 
HankeP  and  M.  Cantor,^  it  may  be  worth  while  to  go  into 
some  detail.^  By  them  it  is  made  to  appear  as  identical 
with  the  procedure  given  later  by  Newton.  The  two  are 
not  the  same.  The  difference  lies  in  the  divisor  used. 
What  is  now  called  '' Newton's  method"  is  Newton's 
method  as  modified  by  Joseph  Raphson.^  The  Newton- 
Raphson  method  of  approximation  to  the  roots  of  an 
equation /(:\;)  =  o  is  usually  given  the  form  a—\f{a)/f'{a)\^ 
where  a  is  an  approximate  value  of  the  required  root. 
It  will  be  seen  that  the  divisor  is /(a).  Vieta's  divisor 
is  different;  it  is 

|/(a+,,)-/(a)  \-s--, 

where/(a;)  is  the  left  of  the  equation/(::t:)  =^,  n  is  the  degree 
of  equation,  and  ^i  is  a  unit  of  the  denomination  of  the 
digit  next  to  be  found.  Thus  in  0:3+420000:^  =  247651713, 
it  can  be  shown  that  417  is  approximately  a  root;  suppose 
that  a  has  been  taken  to  be  400,  then  5i  =  io;  but  if,  at 
the  next  step  of  approximation,  a  is  taken  to  be  410,  then 
5i  =  i.     In  this  example,  taking  ^^  =  400,  Vieta's  divisor 

^  H.  Hankel,  Geschichte  der  Mathematik  in  AUerthum  und  Mit- 
telalter,  Leipzig,  1874,  pp.  369,  370. 

2  M.  Cantor,  Vorlesungen  iiher  Geschichte  der  Mathematik,  II, 
1900,  pp.  640,  641. 

3  This  matter  has  been  discussed  in  a  paper  by  F.  Cajori,  "A 
History  of  the  Arithmetical  Methods  of  Approximation,  etc., 
Colorado  College  Publication,  General  Series  No.  51,  19 10,  pp.  182-84. 
Later  this  subject  was  again  treated  by  G.  Enestrom  in  Bibliotheca 
mathematica,  3.  Folge,  Vol.  XI,  191 1,  pp.  234,  235. 

4  See  F.  Cajori,  op.  cit.,  p.  193. 


Principal  Works  41 

would  have  been  9120000;  Newton's  divisor  would  have 
been  900000. 

A  comparison  of  Vieta's  method  with  the  Newton- 
Raphson  method  reveals  the  fact  that  Vieta's  divisor 
is  more  reliable,  but  labors  under  the  very  great  disad- 
vantage of  requiring  a  much  larger  amount  of  computa- 
tion. The  latter  divisor  is  accurate  enough  and  easier 
to  compute.  Altogether  the  Newton-Raphson  process 
marks  a  decided  advance  over  that  of  Vieta. 

As  already  stated,  it  is  the  method  of  Vieta  that 
Oughtred  explains.  The  Englishman's  exposition  is  an 
improvement  on  that  of  Vieta,  printed  forty  years  earlier. 
Nevertheless,  Oughtred's  explanation  is  far  from  easy 
to  follow.  The  theory  of  equations  was  at  that  time  still 
in  its  primitive  stage  of  development.  Algebraic  nota- 
tion was  not  sufficiently  developed  to  enable  the  argument 
to  be  condensed  into  a  form  easily  surveyed.  So  com- 
plicated does  Vieta's  process  of  approximation  appear 
that  M.  Cantor  failed  to  recognize  that  Vieta  possessed 
a  uniform  mode  of  procedure.  But  when  one  has  in  mind 
the  general  expression  for  Vieta's  divisor  which  we  gave 
above,  one  will  recognize  that  there  was  marked  uni- 
formity in  Vieta's  approximations. 

Oughtred  allows  himself  twenty-eight  sections  in  which 
to  explain  the  process  and  at  the  close  cannot  forbear 
remarking  that  28  is  a  ''perfect"  number  (being  equal 
to  the  sum  of  its  divisors,  i,  2,  4,  7,  14). 

The  early  part  of  his  exposition  shows  how  an  equation 
maybe  transformed  so  as  to  make  its  roots  10, 100, 1000,  or 
10°^  times  smaller.  This  simplifies  the  task  of  "locating  a 
root " ;  that  is,  of  finding  between  what  integers  the  root  lies. 

Taking  one  of  Oughtred's  equations,  x^  —  ']2x^-\-22)^6oox 
=  8725815,  upon  dividing  "jix^  by  10,  238600a:  by  1000, 


42  William  Oughtred 

and  8725815  by  lo^ooo,  we  obtain  x^  —  'j-2x^-\-27,S-6x  = 
872-5.  Dividing  both  sides  by  x,  we  obtain  0:^+238 •  6  — 
7'2a;^  =  x)872'5.  Letting  x  =  4,  we  have  64+238-6  — 
ii5-2  =  i87-4. 

But  4)872 •5(218-1;  4  is  too  small.  Next  let  x  =  5, 
we  have  125  +  238-6  —  180  =  183-6. 

But  5)872-5(174-5;  5  is  too  large.  We  take  the  lesser 
value,  x  =  4,  or  in  the  original  equation,  a;  =  40.  This 
method  may  be  used  to  find  the  second  digit  in  the  root. 
Oughtred  divides  both  sides  of  the  equation  by  x^,  and 
obtains  x^+x)2386oo— 72x  =  :\;^)87258i5.  He  tries  x=4y 
and  ::t:  =  48,  and  finds  that  0^  =  47. 

He  explains  also  how  the  last  computation  may  be 
done  by  logarithms.  Thereby  he  established  for  himself 
the  record  of  being  the  first  to  use  logarithms  in  the  solu- 
tion of  affected  equations. 

As  an  illustration  of  Oughtred's  method  of  approxima- 
tion after  the  root  sought  has  been  located,  we  have 
chosen  for  brevity  a  cubic  in  preference  to  a  quartic.  We 
selected  the  equation  a;3+42oooox  =  2476517 13.  By  the 
process  explained  above  a  root  is  found  to  lie  between 
X  =  400  and  X  =  500.  From  this  point  on,  the  approxima- 
tion as  given  by  Oughtred  is  as  shown  on  p.  43. 

In  further  explanation  of  this  process,  observe  that  the 
given  equation  is  of  the  form  Lc-\-CqL  =  Dc,  where  Lc 
is  our  X,  Q  =  420000,  1)^  =  247651713.  In  the  first  step 
of  approximation,  let  L  =  A-]-E,  where  ^=400  and  E  is, 
as  yet,  undetermined.     We  have 

Lc  =  {A+Ey  =  A^-{-S^"E+3^^"+^' 
and 

CqL = 4  20000  (^  +-E). 

Subtract  from  24765 17 13  the  sum  of  the  known  terms  A^ 


Principal  Works 

"EXEMPLUM   II 
IC+420000/  =  24765 17 13 

Hoc  est,  Lc-\-CqL  =  Dc 


43 


247 

65i 

X  }i. 

(417 

42 

000 

0               Cq 

64 

168 

000 

0 

Ac 

CqA 

232 

000 

0 

Ablatit. 

15 

65i 

713 

4 
4 

8 

I  2 
200 

0  0 

ZAq 

?>A 

9 

120 

0  0 

Divisor. 

4 
4 

8 
I  2 

I 
200 

0  0 

ZAqE 
?>A     Eq 

Ec 

CqE 

9 

121 

0  0 

Ablatit. 

R 


6    530 

713 

504 

I 

420 

3 

23 
000 

?>Aq 

ZA 

92  5 

530 

Divisor. 

3 
2 

530 
60 

940 

I 

27 

343 
000 

3AE 
2>A    Eq 
Ec 

CqE 

6 

530 

7  13 

Ablatit." 

4    I 


16 


16    8     I 


(his  A^  and  420000  A  (his  CqA).     This  sum  is  232000000 
the  remainder  is  15651713. 


44  William  Oughtred 

Next,  he  evaluates  the  coefficients  oi  E  in  t,  A^E  and 
420000  £,  also  3^,  the  coefficient  of  E^.  He  obtains 
3^1^  =  480000,  3^  =  1200,  Cg =420000.  He  interprets 
7,A^  and  Q  as  tens,  3^  as  hundreds.  Accordingly,  he 
obtains  as  their  sum  9120000,  which  is  the  divisor  for 
finding  the  second  digit  in  the  approximation.  Observe 
that  this  divisor  is  the  value  of  \fia-\-Si)—f{a)  \—Si^  in 
our  general  expression,  where  a  =  400,  ,^1  =  10,  n  =  $,f{x)  = 
a;3+42oooox. 

Dividing  the  remainder  15651713  by  9120000,  he  ob- 
tains the  integer  i  in  ten's  place;  thus  E=io,  approxi- 
mately. He  now  computes  the  terms  sA^E,  ^AE^  and 
E^  to  be,  respectively,  4800000,  120000,  1000.  Their 
sum  is  91 2 1000.  Subtracting  it  from  the  previous 
remainder,  1565 17 13,  leaves  the  new  remainder,  6530713. 

From  here  on  each  step  is  a  repetition  of  the  pre- 
ceding step.  The  new  A  is  410,  the  new  E  is  to  be 
determined.  We  have  now  in  closer  approximation, 
L  =  A-\-E.  This  time  we  do  not  subtract  A^  and  CqA, 
because  this  subtraction  is  already  affected  by  the  pre- 
ceding work. 

We  find  the  second  trial  divisor  by  computing  the  sum 
of  3^^,  ^A  and  Q;  that  is,  the  sum  of  504300,  1230, 
420000,  which  is  925530.  Again,  this  divisor  can  be  com- 
puted by  our  general  expression  for  divisors,  by  taking 
a  =  4io,  Si  =  i,  n  =  2>' 

Dividing  6530713  by  925530  yields  the  integer  7 .  Thus 
£=7.  Computing  3^4 ^E,  :^AE^,  E^  and  subtracting  their 
sum,  the  remainder  is  o.  Hence  417  is  an  exact  root  of 
the  given  equation. 

Since  the  extraction  of  a  cube  root  is  merely  the  solu- 
tion of  a  pure  cubic  equation,  x^  =  ny  the  process  given 
above  may  be  utilized  in  finding  cube  roots.     This  is 


Principal  Works  45 

precisely  what  Oughtred  does  in  chap,  xiv  of  his  Clavis. 
If  the  foregoing  computation  is  modified  by  putting  Q  =  o, 
the  process  will  yield  the  approximate  cube  root  of 
2476517 13. 

Oughtred  solves  16  examples  by  the  process  of  approxi- 
mation here  explained.  Of  these,  9  are  cubics,  5  are 
quartics,  and  2  are  quintics.  In  all  cases  he  finds  only 
one  or  two  real  roots.  Of  the  roots  sought,  five  are  irra- 
tional, the  remaining  are  rational  and  are  computed  to 
their  exact  values.  Three  of  the  computed  roots  have  2 
figures  each,  9  roots  have  3  figures  each,  4  roots  have  4 
figures  each.  While  no  attempt  is  made  to  secure  all  the 
roots — methods  of  computing  complex  roots  were  invented 
much  later — he  computes  roots  of  equations  which  involve 
large  coefficients  and  some  of  them  are  of  a  degree  as  high 
as  the  fifth.  In  view  of  the  fact  that  many  editions  of 
the  Clavis  were  issued,  one  impression  as  late  as  1702,  it 
contributed  probably  more  than  any  other  book  to  the 
popularization  of  Vieta's  method  in  England. 

Before  Oughtred,  Thomas  Harriot  and  William  Mil- 
bourn  are  the  only  Englishmen  known  to  have  solved 
numerical  equations  of  higher  degrees.  Milbourn  pub- 
lished nothing.  Harriot  slightly  modified  Vieta's  process 
by  simplifying  somewhat  the  formation  of  the  trial  divisor. 
This  method  of  approximation  was  the  best  in  existence 
in  Europe  until  the  publication  by  Wallis  in  1685  of  New- 
ton's method  of  approximation. 

It  should  be  stated  that,  before  the  time  of  Newton, 
the  best  method  of  approximation  to  the  roots  of  numeri- 
cal equations  existed,  not  in  Europe,  but  in  China.  As 
early  as  the  thirteenth  century  the  Chinese  possessed  a 
method  which  is  almost  identical  with  what  is  known 
today  as  "Horner's  method." 


46  William  Oughtred 

LOGARITHMS 

Oughtred's  treatment  of  logarithms  is  quite  in  accord- 
ance with  the  more  recent  practice.^  He  explains  the 
finding  of  the  ''index"  (our  "characteristic");  he  states 
that  "the  sum  of  two  Logarithms  is  the  Logarithm  of  the 
Product  of  their  Valors;  and  their  difference  is  the 
Logarithm  of  the  Quotient,"  that  "the  Logarithm  of  the 
side  [436]  drawn  upon  the  Index  number  [2]  of  dimensions 
of  any  Potestas  is  the  logarithm  of  the  same  Potestas" 
[436^],  that  "the  logarithm  of  any  Potestas  [436'']  divided 
by  the  number  of  its  dimensions  [2]  affordeth  the  Loga- 
rithm of  its  Root  [436]."  These  statements  of  Oughtred 
occur  for  the  first  time  in  the  Key  of  the  Mathematicks  of 
1647;  the  Clavis  of  163 1  contains  no  treatment  of 
logarithms. 

If  the  characteristic  of  a  logarithm  is  negative,  Ought- 
red indicates  this  fact  by  placing  the  —  above  the  char- 
acteristic. He  separates  the  characteristic  and  mantissa 
by  a  comma,  but  still  uses  the  sign  L  to  indicate  decimal 
fractions.    He  uses  the  contraction  "log." 

INVENTION   OF   THE   SLIDE   RULE;     CONTROVERSY   ON 
PRIORITY   OF   INVENTION 

Oughtred's  most  original  line  of  scientific  activity  is 
the  one  least  known  to  the  present  generation.  Augustus 
De  Morgan,  in  speaking  of  Oughtred,  who  was  sometimes 
called  "Oughtred  Aetonensis,"  remarks:  "He  is  an 
animal  of  extinct  race,  an  Eton  mathematician.  Few 
Eton  men,  even  of  the  minority  which  knows  what  a 
sliding  rule  is,  are  aware  that  the  inventor  was  of  their 

^  See  William  Oughtred's  Key  of  the  Mathematicks,  London,  1694, 
pp.  173-75,  tract,  "Of  the  Resolution  of  the  Affected  Equations," 
or  any  edition  of  the  Clavis  after  the  first. 


Principal  Works  47 

own  school  and  college."^  The  invention  of  the  sHde 
rule  has,  until  recently,^  been  a  matter  of  dispute;  it 
has  been  erroneously  ascribed  to  Edmund  Gunter, 
Edmund  Wingate,  Seth  Partridge,  and  others.  We  have 
been  able  to  establish  that  William  Oughtred  was  the 
first  inventor  of  slide  rules,  though  not  the  first  to  publish 
thereon.  We  shall  see  that  Oughtred  invented  slide 
rules  about  1622,  but  the  descriptions  of  his  instruments 
were  not  put  into  print  before  1632  and  1633.  Meanwhile 
one  of  his  own  pupils,  Richard  Delamain,  who  probably 
invented  the  circular  slide  rule  independently,  published 
a  description  in  1630,  at  London,  in  a  pamphlet  of  32 
pages  entitled  Grammelogia;  or  the  Mathematicall  Ring. 
In  editions  of  this  pamphlet  which  appeared  during  the 
following  three  or  four  years,  various  parts  were  added  on, 
and  some  parts  of  the  first  and  second  editions  eliminated. 
Thus  Delamain  antedates  Oughtred  two  years  in  the 
publication  of  a  description  of  a  circular  slide  rule.  But 
Oughtred  had  invented  also  a  rectilinear  slide  rule,  a 
description  of  which  appeared  in  1633.  To  the  inven- 
tion of  this  Oughtred  has  a  clear  title.  A  bitter  contro- 
versy sprang  up  between  Delamain  on  one  hand,  and 
Oughtred  and  some  of  his  pupils  on  the  other,  on  the 
priority  and  independence  of  invention  of  the  circular  slide 
rule.  Few  inventors  and  scientific  men  are  so  fortunate 
as  to  escape  contests.  The  reader  needs  only  to  recall 
the  disputes  which  have  arisen,  involving  the  researches 
of  Sir  Isaac  Newton  and  Leibniz  on  the  differential  and 
integral  calculus,  of  Thomas  Harriot  and  Rene  Descartes 
relating  to  the  theory  of  equations,  of  Robert  Mayer, 

^  A.  De  Morgan,  op.  ciL,  p.  451;   2d  ed.,  Vol.  II,  p.  303. 
2  See  F.  Cajori,  History  of  the  Logarithmic  Slide  Rule,  New  York, 
1909,  pp.  7-14,  Addenda,  p.  ii. 


48  William  Oughtred 

Hermann  von  Helmholtz,  and  Joule  on  the  principle  of 
the  conservation  of  energy,  or  of  Robert  Morse,  Joseph 
Henry,  Gauss  and  Weber,  and  others  on  the  telegraph, 
to  see  that  questions  of  priority  and  independence  are 
not  uncommon.  The  controversy  between  Oughtred  and 
Delamain  embittered  Oughtred's  life  for  many  years. 
He  refers  to  it  in  print  on  more  than  one  occasion.  We 
shall  confine  ourselves  at  present  to  the  statement  that 
it  is  by  no  means  clear  that  Delamain  stole  the  invention 
from  Oughtred;  Delamain  was  probably  an  independent 
inventor.  Moreover,  it  is  highly  probable  that  the  con- 
troversy would  never  have  arisen,  had  not  some  of  Ought- 
red's  pupils  urged  and  forced  him  into  it.  William  Forster 
stated  in  the  preface  to  the  Circles  of  Proportion  of  1632 
that  while  he  had  been  carefully  preparing  the  manuscript 
for  the  press,  "another  to  whom  the  Author  [Oughtred] 
in  a  louing  confidence  discouered  this  intent,  using  more 
hast  then  good  speed,  went  about  to  preocupate."  It  was 
this  passage  which  started  the  conflagration.  Another 
pupil,  W.  Robinson,  wrote  to  Oughtred,  when  the  latter 
was  preparing  his  Apologeticall  Epistle  as  a  reply  to  Dela- 
main's  countercharges:  ''Good  sir,  let  me  be  beholden 
to  you  for  your  Apology  whensoever  it  comes  forth,  and 
(if  I  speak  not  too  late)  let  me  entreat  you,  whip  ignorance 
well  on  the  blind  side,  and  we  may  turn  him  round,  and 
see  what  part  of  him  is  free."^  As  stated  previously, 
Oughtred's  circular  slide  rule  was  described  by  him  in  his 
Circles  of  Proportion,  London,  1632,  which  was  translated 
from  Oughtred's  Latin  manuscript  and  then  seen  through 
the  press  by  his  pupil,  William  Forster.  In  1633  appeared 
An  Addition  vnto  the  Vse  of  the  Instrvment  called  the  Circles 

^  Rigaud,  op.  cit.,  Vol.  I,  p.  12. 


Principal  Works  49 

of  Proportion  which  contained  at  the  end  "The  Declara- 
tion of  the  two  Rulers  for  Calculation,"  giving  a  descrip- 
tion of  Oughtred's  rectilinear  slide  rule.  This  Addition 
was  bound  with  the  Circles  of  Proportion  as  one  volume. 
About  the  same  time  Oughtred  described  a  modified 
form  of  the  rectilinear  slide  rule,  to  be  used  in  London  for 
gauging.^ 

^  The  New  Artificial  Gauging  Line  or  Rod:  together  with  rules  con- 
cerning the  use  thereof:  Invented  and  written  by  William  Oughtred, 
London,  1633. 


CHAPTER  III 
MINOR  WORKS 

Among  the  minor  works  of  Oughtred  must  be  ranked 
his  booklet  of  forty  pages  to  which  reference  has  already 
been  made,  entitled,  The  New  Artificial  Gauging  Line  or 
Rod  J  London,  1633.  His  different  designs  of  slide  rules 
and  his  inventions  of  sim-dials  as  well  as  his  exposition  of 
the  making  of  watches  show  that  he  displayed  unusual 
interest  and  talent  in  the  various  mathematical  instru- 
ments. A  short  tract  on  watchmaking  was  brought  out 
in  London  as  an  appendix  to  the  Horological  Dialogues 
of  a  clock-  and  watchmaker  who  signed  himself  ''J.  S." 
(John  Smith?).  Oughtred's  tract  appeared  with  its 
own  title-page,  but  with  pagination  continued  from  the 
preceding  part,  ?is  An  Appendix  wherein  is  contained  a 
Method  of  Calculating  all  Numbers  for  Watches.  Written 
originally  by  that  famous  Mathematician  Mr.  William 
Oughtred,  and  now  made  Publick.  By  J.  S.  of  London, 
Clock-maker.    London,  1675. 

"J.  S."  says  in  his  preface: 

The  method  following  was  many  years  since  Compiled  by 
Mr.  Oughtred  for  the  use  of  some  Ingenious  Gentlemen  his 
friends,  who  for  recreation  at  the  University,  studied  to  find 
out  the  reason  and  Knowledge  of  Watch-work,  which  seemed 
also  to  be  a  thing  with  which  Mr.  Oughtred  himself  was  much 
affected,  as  may  in  part  appear  by  his  putting  out  of  his  own 
Son  to  the  same  Trade,  for  whose  use  (as  I  am  informed)  he 
did  compile  a  larger  tract,  but  what  became  of  it  cannot  be 
known. 

50 


Minor  Works  51 

Notwithstanding  Oughtred's  marked  activity  in  the 
design  of  mathematical  instruments,  and  his  use  of  sur- 
veying instruments,  he  always  spoke  in  deprecating  terms 
of  their  importance  and  their  educational  value.  In  his 
epistle  against  Delamain  he  says: 

The  Instruments  I  doe  not  value  or  weigh  one  single  penny. 
If  I  had  been  ambitious  of  praise,  or  had  thought  them  (or  better 
then  they)  worthy,  at  which  to  have  taken  my  rise,  out  of  my 
secure  and  quiet  obscuritie,  to  mount  up  into  glory,  and 
the  knowledge  of  men:  I  could  have  done  it  many  yeares 
before 

Long  agoe,  when  I  was  a  young  student  of  the  Mathemati- 
cal! Sciences,  I  tryed  many  wayes  and  devices  to  fit  my  selve 
with  some  good  DiaU  or  Instrument  portable  for  my  pocket, 
to  finde  the  houre,  and  try  other  conclusions  by,  and  accord- 
ingly framed  for  that  my  purpose  both  Quadrants,  and  Rings, 
and  Cylinders,  and  many  other  composures.  Yet  not  to  my 
full  content  and  satisfaction;  for  either  they  performed  but 
little,  or  els  were  patched  up  with  a  diversity  of  lines  by  an 
unnaturall  and  forced  contexture.  At  last  I  .  .  .  .  found 
what  I  had  before  with  much  studie  and  paines  in  vaine  sought 
for.^ 

Mention  has  been  made  in  the  previous  pages  of  two 
of  his  papers  on  sun-dials,  prepared  (as  he  says)  when  he 
was  in  his  twenty-third  year.  The  first  was  published 
in  the  Clavis  of  1647.  The  second  paper  appeared  in  his 
Circle^  of  Proportion. 

Both  before  and  after  the  time  of  Oughtred  much  was 
written  on  sun-dials.  Such  instruments  were  set  up 
against  the  walls  of  prominent  buildings,  much  as  the 
faces  of  clocks  in  our  time.  The  inscriptions  that  were 
put  upon  sun-dials  are  often  very  clever:  "I  count  only 
the  hours  of  sunshine,"  '^Alas,  how  fleeting."     A  sun-dial 

^  W.  Oughtred,  Apologeticall  Epistle,  p.  13. 


52  William  Oughtred 

on  the  grounds  of  Merchiston  Castle,  in  Edinburgh,  where 
the  inventor  of  logarithms,  John  Napier,  lived  for  many 
years,  bears  the  inscription,  "Ere  time  be  tint,  tak  tent 
of  time"  (Ere  time  be  lost,  take  heed  of  time). 

Portable  sun-dials  were  sometimes  carried  in  pockets,  as 
we  carry  watches.  Thus  Shakespeare,  in  As  You  Like  It, 
Act  II,  sc.  vii: 

"And  then  he  drew  a  diall  from  his  poke." 

Watches  were  first  made  for  carrying  in  the  pocket 
about  1658. 

Because  of  this  literary,  scientific,  and  practical  interest 
in  methods  of  indicating  time  it  is  not  surprising  that 
Oughtred  devoted  himself  to  the  mastery  and  the  advance- 
ment of  methods  of  time-measurement. 

Besides  the  accounts  previously  noted,  there  came 
from  his  pen:  The  Description  and  Use  of  the  double 
Horizontall  Dyall:  Whereby  not  onely  the  hower  of  the  day 
is  shewne;  but  also  the  Meridian  Line  is  found:  And  most 
Astronomical  Questions,  which  may  be  done  by  the  Globe, 
are  resolved.  Invented  and  written  by  W.  O,,  London, 
1636. 

The  "HorizontaU  Dyall"  and  " Horologicall  Ring" 
appeared  again  as  appendixes  to  Oughtred's  translation 
from  the  French  of  a  book  on  mathematical  recreations. 

The  fourth  French  edition  of  that  work  appeared  in 
1627  at  Paris,  under  the  title  of  Recreations  mathematiqve, 
written  by  "Henry  van  Etten,"  a  pseudonym  for  the 
French  Jesuit  Jean  Leurechon  (1591-1690).  English 
editions  appeared  in  1633,  1653,  and  1674.  The  full  title 
of  the  1653  edition  conveys  an  idea  of  the  contents  of  the 
text:  Mathematical  Recreations,  or,  A  Collection  of  many 
Problemes,  extracted  out  of  the  Ancient  and  Modern  Philos- 


Minor  Works  53 

ophers,  as  Secrets  and  Experiments  in  Arithmetick, 
Geometry,  Cosmographie,  Horologiographie,  Astronomie, 
Navigation,  Musick,  Opticks,  Architecture,  Statick,  Me- 
chanicks,  Chemistry,  Water-works,  Fire-works,  b'c.  Not 
vulgarly  manifest  till  now.  Written  first  in  Greek  and 
Latin,  lately  compiVd  in  French,  by  Henry  Van  Etten, 
and  now  in  English,  with  the  Examinations  and  Aug- 
mentations of  divers  Modern  Mathematicians.  W here- 
unto is  added  the  Description  and  Use  of  the  Generall 
Horologicall  Ring.  And  The  Double  Horizontall  Diall. 
Invented  and  written  by  William  Oughtred.  London, 
Printed  for  William  Leake,  at  the  Signe  of  the  Crown  in 
Fleet-street,  between  the  two  Temple-Gates.     MDCLIII. 

The  graphic  solution  of  spherical  triangles  by  the  accu- 
rate drawing  of  the  triangles  on  a  sphere  and  the  measure- 
ment of  the  unknown  parts  in  the  drawing  was  explained 
by  Oughtred  in  a  short  tract  which  was  published  by  his 
son-in-law,  Christopher  Brookes,  under  the  following 
title:  The  Solution  of  all  Sphaerical  Triangles  both  right 
and  oblique  By  the  Planisphaere:  Whereby  two  of  the 
Sphaerical  partes  sought,  are  at  one  position  most  easily 
found  out.  Published  with  consent  of  the  Author,  By 
Christopher  Brookes,  Mathematique  Instrument-maker,  and 
Manciple  of  Wadham  Colledge,  in  Oxford. 

Brookes  says  in  the  preface: 

I  have  oftentimes  seen  my  Reverend  friend  Mr.  W.  O. 
in  his  resolution  of  aU  sphaericaU  triangles  both  right  and 
oblique,  to  use  a  planisphaere,  without  the  tedious  labour  of 
Trigonometry  by  the  ordinary  Canons:  which  planisphaere 
he  had  delineated  with  his  own  hands,  and  used  in  his  calcula- 
tions more  than  Forty  years  before. 

Interesting  as  one  of  our  sources  from  which  Oughtred 
obtained  his  knowledge  of  the  conic  sections  is  his  study 


54  William  Oughtred 

of  Mydorge.  A  tract  which  he  wrote  thereon  was  pub- 
lished by  Jonas  Moore,  in  his  Arithmetick  in  two  hooks 
....  [containing  also]  the  two  first  hooks  of  Mydorgius  his 
conical  sections  analyzed  hy  that  reverend  devine  Mr.  W. 
Oughtred,  Englished  and  completed  with  cuts.  London, 
1660.    Another  edition  bears  the  date  1688. 

To  be  noted  among  the  minor  works  of  Oughtred  are 
his  posthumous  papers.  He  left  a  considerable  number 
of  mathematical  papers  which  his  friend  Sir  Charles 
Scarborough  had  revised  under  his  direction  and  published 
at  Oxford  in  1676  in  one  volume  under  the  title,  Gulielmi 
Oughtredi,  Etonensis,  quondam  Collegii  Regalis  in  Canta- 
brigia  Socii,  Opuscula  Mathematica  hactenus  inedita.  Its 
nine  tracts  are  of  little  interest  to  a  modern  reader. 

Here  we  wish  to  give  our  reasons  for  our  belief  that 
Oughtred  is  the  author  of  an  anonymous  tract  on  the  use 
of  logarithms  and  on  a  method  of  logarithmic  interpolation 
which,  as  previously  noted,  appeared  as  an  ''Appendix" 
to  Edward  Wright's  translation  into  English  of  John 
Napier's  Descriptio,  under  the  title,  A  Description  of  the 
Admirable  Table  of  Logarithmes,  London,  16 18.  The 
"Appendix"  bears  the  title,  ''An  Appendix  to  the  Loga- 
rithmes, showing  the  practise  of  the  Calculation  of  Tri- 
angles, and  also  a  new  and  ready  way  for  the  exact  finding 
out  of  such  lines  and  Logarithmes  as  are  not  precisely 
to  be  found  in  the  Canons."  It  is  an  able  tract.  A 
natural  guess  is  that  the  editor  of  the  book,  Samuel  Wright, 
a  son  of  Edward  Wright,  composed  this  "Appendix." 
More  probable  is  the  conjecture  which  (Dr.  J.  W.  L. 
Glaisher  informs  me)  was  made  by  Augustus  De  Morgan, 
attributing  the  authorship  to  Oughtred.  Two  reasons 
in  support  of  this  are  advanced  by  Dr.  Glaisher,  the  use  of 
X  in  the  "Appendix"  as  the  sign  of  multiplication  (to 


Minor  Works  55 

Oughtred  is  generally  attributed  the  introduction  of  the 
cross  X  for  multiplication  in  1631),  and  the  then  unusual 
designation  "cathetus"  for  the  vertical  leg  of  a  right 
triangle,  a  term  appearing  in  Oughtred's  books.  We  are 
able  to  advance  a  third  argument,  namely,  the  occurrence 
in  the  ''Appendix"  of  (5*)  as  the  notation  for  sine  com- 
plement (cosine),  while  Seth  Ward,  an  early  pupil  of 
Oughtred,  in  his  Idea  trigonometriae  demonstratae,  Oxford, 
1654,  used  a  similar  notation  (5').  It  has  been  stated 
elsewhere  that  Oughtred  claimed  Seth  Ward's  exposition 
of  trigonometry  as  virtually  his  own.  Attention  should 
be  called  also  to  the  fact  that,  in  his  Trigonometria,  p.  2, 
Oughtred  uses  (')  to  designate  i8o°— angle. 

Dr.  J.  W.  L.  Glaisher  is  the  first  to  call  attention  to 
other  points  of  interest  in  this  "Appendix."  The  inter- 
polations are  effected  with  the  aid  of  a  small  table  contain- 
ing the  logarithms  of  72  sines.  Except  for  the  omission 
of  the  decimal  point,  these  logarithms  are  natural  loga- 
rithms— the  first  of  their  kind  ever  published.  In  this 
table  we  find  log  10  =  2302584;  in  modern  notation,  this 
is  stated,  log^  10  =  2.302584.  The  first  more  extended 
table  of  natural  logarithms  of  numbers  was  published  by 
John  Speidell  in  the  1622  impression  of  his  New  Loga- 
rithmes,  which  contains,  besides  trigonometric  tables,  the 
logarithms  of  the  numbers  i-iooo. 

The  "Appendix"  contains  also  the  first  account  of  a 
method  of  computing  logarithms,  called  the  "radix 
method,"  which  is  usually  attributed  to  Briggs  who 
applied  it  in  his  Arithmetica  logarithmica,  1624.  In 
general,  this  method  consists  in  multiplying  or  dividing 
a  number,  whose  logarithm  is  sought,  by  a  suitable  factor 

and  resolving  the  result  into  factors  of  the  form  i  ±  — . 


56  William  Oughtred 

The  logarithm  of  the  number  is  then  obtained  by  adding 
the  previously  calculated  logarithms  of  the  factors.  The 
method  has  been  repeatedly  rediscovered,  by  Flower  in 
177 1,  Atwood  in  1786,  Leonelli  in  1802,  Manning  in  1806, 
Weddle  in  1845,  Hearn  in  1847,  and  Orchard  in  1848. 
We  conclude  with  the  words  of  Dr.  J.  W.  L.  Glaisher: 

The  Appendix  was  an  interesting  and  remarkable  contribu- 
tion to  mathematics,  for  in  its  sixteen  small  pages  it  contains 
(i)  the  first  use  of  the  sign  X ;  (2)  the  first  abbreviations,  or 
symbols,  for  the  sine,  tangent,  cosine,  and  cotangent;  (3)  the 
invention  of  the  radix  method  of  calculating  logarithms; 
(4)  the  first  table  of  hyperbolic  logarithms.^ 

^  Quarterly  Journal  of  Pure  and  Applied  Mathematics,  Vol.  XL VI, 
(1915),  p.  169.  In  this  article  Glaisher  republishes  the  "Appendix" 
in  full. 


CHAPTER  IV 

OUGHTRED'S  INFLUENCE  UPON  MATHEMATICAL 
PROGRESS  AND  TEACHING 

OUGHTRED   AND  HARRIOT 

Oughtred's  Clavis  mathematicae  was  the  most  influential 
mathematical  publication  in  Great  Britain  which  appeared 
in  the  interval  between  John  Napier's  Mirifici  logarith- 
morum  canonis  descriptio,  Edinburgh,  1614,  and  the  time, 
forty  years  later,  when  John  Wallis  began  to  publish 
his  important  researches  at  Oxford.  The  year  163 1  is  of 
interest  as  the  date  of  publication,  not  only  of  Oughtred's 
Clavis y  but  also  of  Thomas  Harriot's  Artis  analyticae 
praxis.  We  have  no  evidence  that  these  two  mathe- 
maticians ever  met.  Through  their  writings  they  did 
not  influence  each  other.  Harriot  died  ten  years  before 
the  appearance  of  his  magnum  opus,  or  ten  years  before 
the  publication  of  Oughtred's  Clavis,  Strangely,  Ought- 
red,  who  survived  Harriot  thirty-nine  years,  never  men- 
tions him.  There  is  no  doubt  that,  of  the  two,  Harriot 
was  the  more  original  mind,  more  capable  of  penetrating 
into  new  fields  of  research.  But  he  had  the  misfortune  of 
having  a  strong  competitor  in  Rene  Descartes  in  the 
development  of  algebra,  so  that  no  single  algebraic 
achievement  stands  out  strongly  and  conspicuously  as 
Harriot's  own  contribution  to  algebraic  science.  As  a 
text  to  serve  as  an  introduction  to  algebra,  Harriot's 
Artis  analyticae  praxis  was  inferior  to  Oughtred's  Clavis. 
The  former  was  a  much  larger  book,  not  as  conveniently 
portable,  compiled  after  the  author's  death  by  others, 

57 


58  William  Oughtred 

and  not  prepared  with  the  care  in  the  development  of  the 
details,  nor  with  the  coherence  and  unity  and  the  profound 
pedagogic  insight  which  distinguish  the  work  of  Oughtred. 
Nor  was  Harrioj:'s  position  in  life  such  as  to  be  surrounded 
by  so  wide  a  circle  of  pupils  as  was  Oughtred.  To  be 
sure,  Harriot  had  such  followers  as  Torporley,  William 
Lower,  and  Protheroe  in  Wales,  but  this  group  is  small  as 
compared  with  Oughtred's. 

OUGHTRED 'S  PUPILS 

There  was  a  large  number  of  distinguished  men 
who,  in  their  youth,  either  visited  Oughtred's  home 
and  studied  under  his  roof  or  else  read  his  Clavis  and 
sought  his  assistance  by  correspondence.  We  permit 
Aubrey  to  enumerate  some  of  these  pupils  in  his  own 
gossipy  style: 

Seth  Ward,  M.A.,  a  fellow  of  Sydney  Colledge  in  Cam- 
bridge (now  bishop  of  Sarum),  came  to  him,  and  lived  with 
him  halfe  a  yeare  (and  he  would  not  take  a  farthing  for  his 
diet),  and  learned  aU  his  mathematiques  of  him.  Sir  Jonas 
More  was  with  him  a  good  while,  and  learn't;  he  was  but  an 
ordinary  logist  before.  Sir  Charles  Scarborough  was  his 
scholar;  so  Dr.  John  Wallis  was  his  scholar;  so  was  Chris- 
topher  Wren   his   scholar,    so   was   Mr Smethwyck, 

Regiae  Societatis  Socius.  One  Mr.  Austin  (a  most  ingeniose 
man)  was  his  scholar,  and  studyed  so  much  that  he  became 
mad,  fell  a  laughing,  and  so  dyed,  to  the  great  griefe  of  the  old 

gentleman.    Mr Stokes,    another    scholar,    feU    mad, 

and  dream't  that  the  good  old  gentleman  came  to  him,  and 
gave  him  good  advice,  and  so  he  recovered,  and  is  still  weU. 
Mr.  Thomas  Henshawe,  Regiae  Societatis  Socius,  was  his 
scholar  (then  a  young  gentleman).  But  he  did  not  so  much 
like  any  as  those  that  tugged  and  tooke  paines  to  worke  out 
questions.    He  taught  all  free. 


Influence  on  Mathematical  Progress        59 

He  could  not  endure  to   see  a  scholar  write  an  iU  hand; 
he  taught  them  all  presently  to  mend  their  hands. ^ 

Had  Oughtred  been  the  means  of  guiding  the  mathe- 
matical studies  of  only  John  Wallis  and  Christopher 
Wren — one  the  greatest  English  mathematician  between 
Napier  and  Newton,  the  other  one  of  the  greatest  archi- 
tects of  England — he  would  have  earned  profound  grati- 
tude. But  the  foregoing  list  embraces  nine  men,  most  of 
them  distinguished  in  their  day.  And  yet  Aubrey's  list 
is  very  incomplete.  It  is  easy  to  more  than  double  it  by 
adding  the  names  of  William  Forster,  who  translated  from 
Latin  into  English  Oughtred's  Circles  of  Proportion;  Arthur 
Haughton,  who  brought  out  the  1660  Oxford  edition  of 
the  Circles  of  Proportion;  Robert  Wood,  an  educator 
and  politician,  who  assisted  Oughtred  in  the  translation 
of  the  Clavis  from  Latin  into  English  for  the  edition 
of  1647;  W.  Gascoigne,  a  man  of  promise,  who  fell 
in  1644  at  Marston  Moor;  John  Twysden,  who  was 
active  as  a  publisher;  William  Sudell,  N.  Ewart,  Richard 
Shuttleworth,  William  Robinson,  and  William  Howard, 
the  son  of  the  Earl  of  Arundel,  for  whose  instruction 
Oughtred  originally  prepared  the  manuscript  treatise 
that  was  published  in  163 1  as  the  Clavis  mathematicae. 

Nor  must  we  overlook  the  names  of  Lawrence  Rooke 
(who  "did  admirably  well  read  in  Gresham  Coll.  on  the 
sixth  chapt.  of  the  said  book,"  the  Clavis);  Christopher 
Brookes  (a  maker  of  mathematical  instruments  who 
married  a  daughter  of  the  famous  mathematician); 
William  Leech  and  William  Brearly  (who  with  Robert 
Wood  ''have  been  ready  and  helpfull  incouragers  of  me 
[Oughtred]  in  this  labour"  of  preparing  the  English  Clavis 

^  Aubrey,  op.  cit.,  Vol.  II,  1898,  p.  ic8. 


6o  William  Oughtred 

of  1647),  and  Thomas  Wharton,  who  studied  the  Clavis 
and  assisted  in  the  editing  of  the  edition  of  1647. 

The  devotion  of  these  pupils  offers  eloquent  testimony, 
not  only  of  Oughtred 's  ability  as  a  mathematician,  but 
also  of  his  power  of  drawing  young  men  to  him — of  his 
personal  magnetism.  Nor  should  we  omit  from  the  list 
Richard  Delamain,  a  teacher  of  mathematics  in  London, 
who  unfortunately  had  a  bitter  controversy  with  Ought- 
red on  the  priority  and  independence  of  the  invention  of 
the  circular  slide  rule  and  a  form  of  sun-dial.  Delamain 
became  later  a  tutor  in  mathematics  to  King  Charles  I, 
and  perished  in  the  civil  war,  before  1645. 

oughtred,  the  '^todhunter  of  the  seventeenth 

century" 

To  afford  a  clearer  view  of  Oughtred  as  a  teacher  and 
mathematical  expositor  we  quote  some  passages  from 
various  writers  and  from  his  correspondence.  Anthony 
Wood^  gives  an  interesting  account  of  how  Seth  Ward 
and  Charles  Scarborough  went  from  Cambridge  Uni- 
versity to  the  obscure  home  of  the  country  mathematician 
to  be  initiated  into  the  mysteries  of  algebra: 

Mr.  Cha.  Scarborough,  then  an  ingenious  young  student 
and  fellow  of  Caius  Coll.  in  the  same  university,  was  his  [Seth 
Ward's]  great  acquaintance,  and  both  being  equally  students 
in  that  faculty  and  desirous  to  perfect  themselves,  they  took 
a  journey  to  Mr.  Will.  Oughtred  living  then  at  Albury  in 
Surrey,  to  be  informed  in  many  things  in  his  Clavis  mathematica 
which  seemed  at  that  time  very  obscure  to  them.  Mr.  Ought- 
red treated  them  with  great  humanity,  being  very  much  pleased 
to  see  such  ingenious  young  men  apply  themselves  to  these 
studies,  and  in  short  time  he  sent  them  away  well  satisfied  in 
their  desires.    When  they  returned  to  C§,mbridge,  they  after- 

^  Wood's  Athenae  Oxonienses  (ed,  P.  Bliss),  Vol.  IV,  1820,. p.  247. 


Influence  on  Mathematical  Progress        6i 

wards  read  the  Clav.  Math,  to  their  pupils,  which  was  the  first 
time  that  book  was  read  in  the  said  university.  Mr.  Laur. 
Rook,  a  disciple  of  Oughtred,  I  think,  and  Mr.  Ward's  friend, 
did  admirably  well  read  in  Gresham  Coll.  on  the  sixth  chap,  of 
the  said  book,  which  obtained  him  great  repute  from  some  and 
greater  from  Mr.  Ward,  who  ever  after  had  an  especial  favour 
for  him. 

Anthony  Wood  makes  a  similar  statement  about 
Thomas  Henshaw: 

While  he  remained  in  that  coll.  [University  CoUege,  Oxford] 
which  was  five  years  ....  he  made  an  excursion  for  about 
9  months  to  the  famous  mathematician  Will.  Oughtred  parson 
of  Aldbury  in  Surrey,  by  whom  he  was  initiated  in  the  study 
of  mathematics,  and  afterwards  retiring  to  his  coll.  for  a  time, 
he  at  length  went  to  London,  was  entered  a  student  in  the 
Middle  Temple.^ 

Extracts  from  letters  of  W.  Gascoigne  to  Oughtred, 
of  the  years  1640  and  1641,  throw  some  light  upon  mathe- 
matical teaching  of  the  time: 

Amongst  the  mathematical  rarities  these  times  have 
afforded,  there  are  none  of  that  small  number  I  (a  late  intruder 
into  these  studies)  have  yet  viewed,  which  so  fully  demonstrates 
their  authors'  great  abihties  as  your  Clavis,  not  richer  in 
augmentations,  than  valuable  for  contraction;  .... 

Your  behef  that  there  is  in  all  inventions  ah  quid  divinum, 
an  infusion  beyond  human  cogitations,  I  am  confident  will 
appear  notably  strengthened,  if  you  please  to  afford  this  truth 
behef,  that  I  entered  upon  these  studies  accidentally  after  I 
betook  myself  to  the  country,  having  never  had  so  much  aid  as 
to  be  taught  addition,  nor  the  discourse  of  an  artist  (having  left 
both  Oxford  and  London  before  I  knew  what  any  proposition  in 
geometry  meant)  to  inform  me  what  were  the  best  authors.^ 

^  Wood,  op.  cit.,  Vol.  II,  p.  445. 
^JRigaud,  op.  cit.,  Vol.  I,  pp.  33,  35. 


62  William  Oughtred 

The  following  extracts  from  two  letters  by  W.  Robin- 
son, written  before  the  appearance  of  the  1647  English 
edition  of  the  Clavis,  express  the  feeling  of  many  readers 
of  the  Clavis  on  its  extreme  conciseness  and  brevity  of 
explanation: 

I  shall  long  exceedingly  tUl  I  see  your  Clavis  turned  into 
a  pick-lock;  and  I  beseech  you  enlarge  it,  and  explain  it  what 
you  can,  for  we  shaU  not  need  to  fear  either  tautology  or  super- 
fluity; you  are  naturally  concise,  and  your  clear  judgment 
makes  you  both  methodical  and  pithy;  and  your  analytical 
way  is  indeed  the  only  way 

I  wiU  once  again  earnestly  entreat  you,  that  you  be  rather 
diffuse  in  the  setting  forth  of  your  EngHsh  mathematical  Clavis, 
than  concise,  considering  that  the  wisest  of  men  noted  of  old, 
and  said  stultorum  infinitus  est  numerus,  these  arts  cannot  be 
made  too  easy,  they  are  so  abstruse  of  themselves,  and  men 
either  so  lazy  or  dull,  that  their  fastidious  wits  take  a  loathing 
at  the  very  entrance  of  these  studies,  unless  it  be  sweetened  on 
with  plainness  and  facility.  Brevity  may  weU  argue  a  learned 
author,  that  without  any  excess  or  redundance,  either  of  matter 
or  words,  can  give  the  very  substance  and  essence  of  the  thing 
treated  of;  but  it  seldom  makes  a  learned  scholar;  and  if  one 
be  capable,  twenty  are  not;  and  if  the  master  sum  up  in  brief 
the  pith  of  his  own  long  labours  and  travails,  it  is  not  easy  to 
imagine  that  scholars  can  with  less  labour  than  it  cost  their 
masters  dive  into  the  depths  thereof.^ 

Here  is  the  judgment  of  another  of  Oughtred's  friends: 

....  with  the  character  I  received  from  your  and  my  noble 
friend  Sir  Charles  Cavendish,  then  at  Paris,  of  your  second 
edition  of  the  same  piece,  made  me  at  my  return  into  England 
speedily  to  get,  and  diligently  peruse  the  same.  Neither 
truly  did  I  find  my  expectation  deceived;  having  with  admira- 
tion often  considered  how  it  was  possible  (even  in  the  hardest 

I  Rigaud,  op.  ciL,  Vol.  I,  pp.  16,  26. 


Influence  on  Mathematical  Progress        63 

things  of  geometry)  to  deliver  so  much  matter  in  so  few  words, 
yet  with  such  demonstrative  clearness  and  perspicuity:  and 
hath  often  put  me  in  mind  of  learned  Mersennus  his  judgment 
(since  dead)  of  it,  that  there  was  more  matter  comprehended  in 
that  httle  book  than  in  Diophantus,  and  all  the  ancients ^ 

Oughtred's  own  feeling  was  against  diffuseness  in  text- 
book writing.  In  his  revisions  of  his  Clavis  the  original 
character  of  that  book  was  not  altered.  In  his  reply  to 
W.  Robinson,  Oughtred  said : 

.  .  .  .But  my  art  for  all  such  mathematical  inventions  I 
have  set  down  in  my  Clavis  Mathematica,  which  therefore 
in  my  title  I  say  is  tum  logisticae  cum  analyticae  adeoque 
totius  mathematicae  quasi  clavis,  which  if  any  one  of  a. mathe- 
matical genius  wiU  carefully  study,  (and  indeed  it  must  be 
carefully  studied,)  he  will  not  admire  others,  but  himself  do 
wonders.  But  I  (such  is  my  tenuity)  have  enough  fungi  vice 
cotis,  acutum  reddere  quae  ferrum  valet,  exsors  ipsa  secandi, 
or  like  the  touchstone,  which  being  but  a  stone,  base  and  little 
worth,  can  shew  the  excellence  and  riches  of  gold.^ 

John  Wallis  held  Oughtred's  Clavis  in  high  regard. 
When  in  correspondence  with  John  Collins  concerning 
plans  for  a  new  edition,  Wallis  wrote  in  1666-67,  six 
years  after  the  death  of  Oughtred: 

....  But  for  the  goodness  of  the  book  in  itself,  it  is  that 
(I  confess)  which  I  look  upon  as  a  very  good  book,  and  which 
doth  in  as  little  room  deHver  as  much  of  the  fundamental  and 
useful  part  of  geometry  (as  well  as  of  arithmetic  and  algebra) 
as  any  book  I  know;  and  why  it  should  not  be  now  acceptable 
I  do  not  see.  It  is  true,  that  as  in  other  things  so  in  mathe- 
matics, fashions  wiU  daily  alter,  and  that  which  Mr.  Oughtred 
designed  by  great  letters  may  be  now  by  others  be  designed  by 
small;  but  a  mathematician  wUl,  with  the  same  ease  and  ad- 
vantage, understand  Ac,  and  a^  or  aaa And  the  like 

^  Rigaud,  op.  cit.,  Vol.  I,  p.  66.  ^  Ihid.,  Vol.  I,  p.  9. 


64  William  Oughtred 

I  judge  of  Mr.  Oughtred's  Clavis,  which  I  look  upon  (as  those 
pieces  of  Vieta  who  first  went  in  that  way)  as  lasting  books  and 
classic  authors  in  this  kind;  to  which,  notwithstanding,  every 
day  may  make  new  additions.  .... 

But  I  confess,  as  to  my  own  judgment,  I  am  not  for  making 
the  book  bigger,  because  it  is  contrary  to  the  design  of  it,  being 
intended  for  a  manual  or  contract;    whereas  comments,  by 

enlarging  it,  do  rather  destroy  it But  it  was  by  him 

intended,  in  a  small  epitome,  to  give  the  substance  of  what  is 
by  others  delivered  in  larger  volumes ^ 

That  there  continued  to  be  a  group  of  students  and 
teachers  who  desired  a  fuller  exposition  than  is  given  by 
Oughtred  is  evident  from  the  appearance,  over  fifty 
years  after  the  first  publication  of  the  Clavis ,  of  a  booklet 
by  Gilbert  Clark,  entitled  Oughtredus  Explicatus,  London, 
1682.  A  review  of  this  appeared  in  the  Acta  Eruditorum 
(Leipzig,  1684),  on  p.  168,  wherein  Oughtred  is  named 
"clarissimus  Angliae  mathematicus."  John  Collins  wrote 
Wallis  in  1666-67  that  Clark,  ''who  lives  with  Sir  Jus- 
tinian Isham,  within  seven  miles  of  Northampton,  .... 
intimates  he  wrote  a  comment  on  the  Clavis,  which  lay 
long  in  the  hands  of  a  printer,  by  whom  he  was  abused, 
meaning  Leybourne.  "^ 

We  shall  have  occasion  below  to  refer  to  Oughtred's 
inability  to  secure  a  copy  of  a  noted  Italian  mathematical 
work  published  a  few  years  before.  In  those  days  the 
condition  of  the  book  trade  in  England  must  have  been 
somewhat  extraordinary.  Dr.  J.  W.  L.  Glaisher  throws 
some  light  upon  this  subject.^    He  found  in  the  Calendar 

^  Rigaud,  op.  ciL,  Vol.  II,  p.  475.         ''Ibid.,  Vol.  II,  p.  471. 

3  J.  W.  L.  Glaisher,  "On  Early  Logarithmic  Tables,  and  Their 
Calculators,"  Philosophical  Magazine,  4th  Ser.,  Vol.  XLV  (1873), 
PP-  378,  379. 


Influence  on  Mathematical  Progress        65 

of  State  Papers,  Domestic  Series,  1637,  a  petition  to  Arch- 
bishop Laud  in  which  it  is  set  forth  that  when  Hoogan- 
huysen,  a  Dutchman,  '' heretofore  complained  of  in  the 
High  Commission  for  importing  books  printed  beyond 
the  seas,"  had  been  bound  "not  to  bring  in  any  more," 
one  Vlacq  (the  computer  and  pubhsher  of  logarithmic 
tables)  ''kept  up  the  same  agency  and  sold  books  in  his 

stead Vlacq  is  now  preparing  to  go  beyond  the 

seas  to  avoid  answering  his  late  bringing  over  nine  bales  of 
books  contrary  to  the  decree  of  the  Star  Chamber."  Judg- 
ment was  passed  that,  "Considering  the  ill-consequence  and 
scandal  that  would  arise  by  strangers  importing  and  vent- 
ing in  this  kingdom  books  printed  beyond  the  seas,"  certain 
importations  be  prohibited,  and  seized  if  brought  over. 

This  want  of  easy  intercommunication  of  results  of 
scientific  research  in  Oughtred's  time  is  revealed  in  the 
following  letter,  written  by  Oughtred  to  Robert  Keylway, 
in  1645: 

I  speak  this  the  rather,  and  am  induced  to  a  better  con- 
fidence of  your  performance,  by  reason  of  a  geometric-analytical 
art  or  practice  found  out  by  one  Cavalieri,  an  Italian,  of  which 
about  three  years  since  I  received  information  by  a  letter  from 
Paris,  wherein  was  praelibated  only  a  small  taste  thereof,  yet 
so  that  I  divine  great  enlargement  of  the  bounds  of  the  mathe- 
matical empire  wiU  ensue.  I  was  then  very  desirous  to  see  the 
author's  own  book  while  my  spirits  were  more  free  and  light- 
some, but  I  could  not  get  it  in  France.  Since,  being  more  stept 
into  years,  daunted  and  broken  with  the  sufferings  of  these 
disastrous  times,  I  must  content  myself  to  keep  home,  and  not 
put  out  to  any  foreign  discoveries.^ 

It  was  in  1655,  when  Oughtred  was  about  eighty  years 
old,  that  John  Wallis,  the  great  forerunner  of  Newton  in 

^  Rigaud,  op.  cit.,  Vol.  I,  p.  65. 


66  William  Oughtred 

Great  Britain,  began  to  publish  his  great  researches  on 
the  arithmetic  of  infinites.  Oughtred  rejoiced  over  the 
achievements  of  his  former  pupil.  In  1655,  Oughtred 
wrote  John  Wallis  as  follows: 

I  have  with  unspeakable  delight,  so  far  as  my  necessary 
businesses,  the  infirmness  of  my  health,  and  the  greatness  of 
my  age  (approaching  now  to  an  end)  would  permit,  perused 
your  most  learned  papers,  of  several  choice  arguments,  which 
you  sent  me :  wherein  I  do  first  with  thankfulness  acknowledge 
to  God,  the  Father  of  lights,  the  great  light  he  hath  given  you; 
and  next  I  congratulate  you,  even  with  admiration,  the  clear- 
ness and  perspicacity  of  your  understanding  and  genius,  who 
have  not  only  gone,  but  also  opened  a  way  into  these  profound- 
est  mysteries  of  art,  unknown  and  not  thought  of  by  the 
ancients.  With  which  your  mysterious  inventions  I  am  the 
more  affected,  because  full  twenty  years  ago,  the  learned  patron 
of  learning,  Sir  Charles  Cavendish,  shewed  me  a  paper  written, 
wherein  were  some  few  excellent  new  theorems,  wrought  by 
the  way,  as  I  suppose,  of  Cavaheri,  which  I  wrought  over 
again  more  agreeably  to  my  way.  The  paper,  wherein  I 
wrought  it,  I  shewed  to  many,  whereof  some  took  copies,  but 
my  own  I  cannot  find.  I  mention  it  for  this,  because  I  saw 
therein  a  light  breaking  out  for  the  discovery  of  wonders  to 
be  revealed  to  mankind,  in  this  last  age  of  the  world:  which 
light  I  did  salute  as  afar  off,  and  now  at  a  nearer  distance 
embrace  in  your  prosperous  beginnings.  Sir,  that  you  are 
pleased  to  mention  my  name  in  your  never  dying  papers,  that 
is  your  noble  favour  to  me,  who  can  add  nothing  to  your  glory, 
but  only  my  applause ^ 

The  last  sentence  has  reference  to  Wallis'  appreciative 
and  eulogistic  reference  to  Oughtred  in  the  preface.  It 
is  of  interest  to  secure  the  opinion  of  later  English  writers 
who    knew    Oughtred    only    through   his    books.    John 

^  Rigaud,  op.  ciL,  Vol.  I,  p.  87. 


Influence  on  Mathematical  Progress        67 

Locke  wrote  in  his  journal  under  the  date,  June  24,  1681, 
''the  best  algebra  yet  extant  is  Outred's."^  John  Collins, 
who  is  known  in  the  history  of  mathematics  chiefly 
through  his  very  extensive  correspondence  with  nearly 
all  mathematicians  of  his  day,  was  inclined  to  be  more 
critical.    He  wrote  Wallis  about  1667: 

It  was  not  my  intent  to  disparage  the  author,  though  I 
know  many  that  did  Hghtly  esteem  him  when  living,  some 

whereof  are  at  rest,  as  Mr.  Foster  and  Mr.  Gibson 

You  grant  the  author  is  brief,  and  therefore  obscure,  and  I 
say  it  is  but  a  collection,  which,  if  himself  knew,  he  had  done 
well  to  have  quoted  his  authors,  whereto  the  reader  might  have 
repaired.  You  do  not  like  those  words  of  Vieta  in  his  theorems, 
ex  adjunctione  piano  solidi,  plus  quadrato  quadrati,  etc.,  and 
think  Mr.  Oughtred  the  first  that  abridged  those  expressions 
by  symbols;  but  I  dissent,  and  tell  you  'twas  done  before  by 
Cataldus,  Geysius,  and  Camillus  Gloriosus,^  who  in  his  first 
decade  of  exercises,  (not  the  first  tract,)  printed  at  Naples  in 
1627,  which  was  four  years  before  the  first  edition  of  the  Clavis, 
proposeth  this  equation  just  as  I  here  give  it  you,  viz.  iccc-\- 
i6qcc+4iqqc—2304cc—  18364^^-  i33ooo^g-  54505^+3728^+ 
8064  N  aequatur  4608,  finds  iV  or  a  root  of  it  to  be  24,  and  com- 
poseth  the  whole  out  of  it  for  proof,  just  in  Mr.  Oughtred's 
symbols  and  method.  Cataldus  on  Vieta  came  out  fifteen 
years  before,  and  I  cannot  quote  that,  as  not  having  it 
by  me. 

....  And  as  for  Mr.  Oughtred's  method  of  symbols, 
this  I  say  to  it;  it  may  be  proper  for  you  as  a  commentator  to 
follow  it,  but  divers  I  know,  men  of  inferior  rank  that  have  good 

skill  in  algebra,  that  neither  use  nor  approve  it Is  not 

A^  sooner  wrote  than  Aqc?  Let  ^  be  2,  the  cube  of  2  is  8, 
which  squared  is  64:    one  of  the  questions  between  Maghet 

^  King's  Life  of  John  Locke,  Vol.  I,  London,  1830,  p,  227. 

2  Exercitationum  Mathematicarum  Decas  prima,  Naples,  1627,  and 
probably  Cataldus'  Transformatio  Geometrica,  Bonon.,  161 2. 


68  William  Oughtred 

Grisio  and  Gloriosus  is  whether  6^  — Ace  or  Aqc.  The  Cartesian 
method  tells  you  it  is  A^,  and  decides  the  doubt ^ 

There  is  some  ground  for  the  criticisms  passed  by 
Collins.  To  be  sure,  the  first  edition  of  the  Clavis  is 
dated  163 1 — six  years  before  Descartes  suggested  the 
exponential  notation  which  came  to  be  adopted  as  the 
symbolism  in  our  modern  algebra.  But  the  second  edition 
of  the  Clavis,  1647,  appeared  ten  years  after  Descartes' 
innovation.  Had  Oughtred  seen  fit  to  adopt  the  new  expo- 
nential notation  in  1647,  the  step  would  have  been  epoch- 
making  in  the  teaching  of  algebra  in  England.  We  have 
seen  no  indication  that  Oughtred  was  familiar  with  Des- 
cartes' Geometrie  of  1637. 

The  year  preceding  Oughtred's  death  Mr.  John  Twys- 
den  expressed  himself  as  follows  in  the  preface  to  his 
Miscellanies: 

It  remains  that  I  should  adde  something  touching  the  begin- 
ning, and  use  of  these  Sciences I  shaU  only,  to  their 

honours,  name  some  of  our  own  Nation  yet  living,  who  have 
happily  laboured  upon  both  stages.  That  succeeding  ages 
may  understand  that  in  this  of  ours,  there  yet  remained  some 
who  were  neither  ignorant  of  these  Arts,  as  if  they  had  held 
them  vain,  nor  condemn  them  as  superfluous.  Amongst 
them  aU  let  Mr.  William  Oughtred,  of  Aeton,  be  named  in  the 
first  place,  a  Person  of  venerable  grey  haires,  and  exemplary 
piety,  who  indeed  exceeds  all  praise  we  can  bestow  upon 
him.  Who  by  an  easie  method,  and  admirable  Key,  hath 
unlocked  the  hidden  things  of  geometry.  Who  by  an  accu- 
rate Trigonometry  and  furniture  of  Instruments,  hath  in- 
riched,  as  well  geometry,  as  Astronomy.  Let  D.  John  Wallis, 
and  D.  Seth  Ward,  succeed  in  the  next  place,  both  famous 
Persons,  and  Doctors  in  Divinity,  the  one  of  geometry,  the 

^  Rigaud,  op.  cit.,  Vol.  II,  pp.  477-80 


Influence  on  Mathematical  Progress        69 

other   of   astronomy,    Savilian   Professors   in   the  University 
of   Oxford.^ 

The  astronomer  Edmund  Halley,  in  his  preface  to  the 
1694  English  edition  of  the  Clavis,  speaks  of  this  book  as 
one  of  "so  established  a  reputation,  that  it  were  needless 
to  say  anything  thereof,"  though  "the  concise  Brevity 
of  the  author  is  such,  as  in  many  places  to  need  Explica- 
tion, to  render  it  Intelligible  to  the  less  knowing  Mathe- 
matical matters." 

In  closing  this  part  of  our  monograph,  we  quote  the 
testimony  of  Robert  Boyle,  the  experimental  physicist, 
as  given  May  8,  1647,  in  a  letter  to  Mr.  Hartlib: 

The  Englishing  of,  and  additions  to  Oughtred's  Clavis 
mathematica  does  much  content  me,  I  having  formerly  spent 
much  study  on  the  original  of  that  algebra,  which  I  have  long 
since  esteemed  a  much  more  instructive  way  of  logic,  than  that 
of  Aristotle.2 

WAS  DESCARTES  INDEBTED  TO  OUGHTRED  ? 

This  question  first  arose  in  the  seventeenth  century, 
when  John  Wallis,  of  Oxford,  in  his  Algebra  (the  English 
edition  of  1685,  and  more  particularly  the  Latin  edition 
of  1693),  raised  the  issue  of  Descartes'  indebtedness  to  the 
English  scientists,  Thomas  Harriot  and  William  Oughtred. 
In  discussing  matters  of  priority  between  Harriot  and 
Descartes,  relating  to  the  theory  of  equations,  Wallis 
is  generally  held  to  have  shown  marked  partiality  to 
Harriot.    Less  attention  has  been  given  by  historians 

^  Miscellanies:  or  Mathematical  Lucubrations,  of  Mr.  Samuel 
Foster,  Sometimes  publike  Professor  of  Astronomic  in  Gresham  Colledge 
in  London,  by  John  Twysden,  London,  1659. 

^  The  Works  of  the  Honourable  Robert  Boyle  in  five  volumes,  to 
which  is  prefixed  the  Life  of  the  Author,  Vol.  I,  London,  1744,  p.  24. 


70  William  Oughtred 

of  mathematics  to  Descartes'  indebtedness  to  Oughtred. 
Yet  this  question  is  of  importance  in  tracing  Oughtred's 
influence  upon  his  time. 

On  January  8,  1688-89,  Samuel  Morland  addressed  a 
letter  of  inquiry  to  John  Wallis,  containing  a  passage 
which  we  translate  from  the  Latin: 

Some  time  ago  I  read  in  the  elegant  and  truly  precious  book 
that  you  have  written  on  Algebra,  about  Descartes,  this  philos- 
opher so  extolled  above  aU  for  having  arrived  at  a  very  perfect 
system  by  his  own  powers,  without  the  aid  of  others,  this 
Descartes,  I  say,  who  has  received  in  geometry  very  great  light 
from  our  Oughtred  and  our  Harriot,  and  has  followed  their 
track  though  he  carefully  suppressed  their  names.  I  stated 
this  in  a  conversation  with  a  professor  in  Utrecht  (where  I 
reside  at  present).  He  requested  me  to  indicate  to  him  the 
page-numbers  in  the  two  authors  which  justified  this  accusa- 
tion. I  admitted  that  I  could  not  do  so.  The  Geometrie  of 
Descartes  is  not  sufficiently  familiar  to  me,  although  with 
Oughtred  I  am  fairly  familiar.  I  pray  you  therefore  that  you 
wiU  assiune  this  burden.  Give  me  at  least  those  references 
to  passages  of  the  two  authors  from  the  comparison  of  which 
the  plagiarism  by  Descartes  is  the  most  striking.^ 

Following  norland's  letter  in  the  De  algebra  tractatus, 
is  printed  Wallis'  reply,  dated  March  12,  1688  ("Stilo 
Angliae"),  which  is,  in  part,  as  follows: 

I  nowhere  give  him  the  name  of  a  plagiarist;  I  would  not 
appear  so  impoHte.  However  this  I  say,  the  major  part  of  his 
algebra  (if  not  all)  is  found  before  him  in  other  authors  (notably 
in  our  Harriot)  whom  he  does  not  designate  by  name.  That 
algebra  may  be  appHed  to  geometry,  and  that  it  is  in  fact  so 
applied,  is  nothing  new.  Passing  the  ancients  in  silence,  we 
state  that  this  has  been  done  by  Vieta,  Ghetaldi,  Oughtred 

^  The  letter  is  printed  in  John  Wallis'  De  algebra  tractatus,  1693, 
p.  206. 


Influence  on  Mathematical  Progress        71 

and  others,  before  Descartes.     They  have  resolved  by  algebra 
and  specious  arithmetic  [Hteral  arithmetic]  many  geometrical 

problems But  the  question  is  not  as  to  application  of 

algebra  to  geometry  (a  thing  quite  old),  but  of  the  Cartesian 
algebra  considered  by  itself. 

Wallis  then  indicates  in  the  1659  edition  of  Descartes' 
Geometrie  where  the  subjects  treated  on  the  first  six  pages 
are  found  in  the  writings  of  earlier  algebraists,  particu- 
larly of  Harriot  and  Oughtred.  For  example,  what  is 
found  on  the  first  page  of  Descartes,  relating  to  addition, 
subtraction,  multiplication,  division,  and  root  extraction, 
is  declared  by  Wallis  to  be  drawn  from  Vieta,  Ghetaldi, 
and  Oughtred. 

It  is  true  that  Descartes  makes  no  mention  of  modern 
writers,  except  once  of  Cardan.  But  it  was  not  the  pur- 
pose of  Descartes  to  write  a  history  of  algebra.  To  be 
sure,  references  to  such  of  his  immediate  predecessors  as 
he  had  read  would  not  have  been  out  of  place.  Neverthe- 
less, Wallis  fails  to  show  that  Descartes  made  illegiti- 
mate use  of  anything  he  may  have  seen  in  Harriot  or 
Oughtred. 

The  first  inquiry  to  be  made  is,  Did  Descartes  possess 
copies  of  the  books  of  Harriot  and  Oughtred  ?  It  is  only 
in  recent  time  that  this  question  has  been  answered  as  to 
Harriot.  As  to  Oughtred,  it  is  still  unanswered.  It  is 
now  known  that  Descartes  had  seen  Harriot's  Artis  analy- 
ticae  praxis  (163 1).  Descartes  wrote  a  letter  to  Con- 
stantin  Huygens  in  which  he  states  that  he  is  sending 
Harriot's  book.^ 

An  able  discussion  of  the  question,  what  effect,  if 
any,  Oughtred's  Clavis  mathematicae  of  163 1  had  upon 

^  See  La  Correspondance  de  Descartes,  published  by  Charles  Adam 
and  Paul  Tannery,  Vol.  II,  Paris,  1898,  pp.  456  and  457. 


72  William  Oughtred 

Descartes'^  Geometrie  of  1637,  is  given  by  H.  Bosmans  in 
a  recent  article.  According  to  Bosmans  no  evidence  has 
been  found  that  Descartes  possessed  a  copy  of  Oughtred's 
book,  or  that  he  had  examined  it.  Bosmans  beUeves 
nevertheless  that  Descartes  was  influenced  by  the  Clavis, 
either  directly  or  indirectly.     He  says: 

If  Descartes  did  not  read  it  carefully,  which  is  not  proved, 
he  was  none  the  less  well  informed  with  regard  to  it.  No 
one  denies  his  intimate  knowledge  of  the  intellectual  move- 
ment of  his  time.  The  Clavis  mathematica  enjoyed  a  rapid 
success.  It  is  impossible  that,  at  least  indirectly,  he  did  not 
know  the  more  original  ideas  which  it  contained.  Far  from 
belittling  Descartes,  as  I  much  desire  to  repeat,  this  rather 
makes  him  the  greater. ^ 

We  ourselves  would  hardly  go  as  far  as  does  Bosmans. 
Unless  Descartes  actually  examined  a  copy  of  Oughtred 
it  is  not  likely  that  he  was  influenced  by  Oughtred  in 
appreciable  degree.  Book  reviews  were  quite  unknown 
in  those  days.  No  evidence  has  yet  been  adduced  to  show 
that  Descartes  obtained  a  knowledge  of  Oughtred  by 
correspondence.  A  most  striking  feature  about  Ought- 
red's  Clavis  is  its  notation.  No  trace  of  the  Englishman's 
symbolism  has  been  pointed  out  in  Descartes'  Geometrie 
of  1637.  Only  six  years  intervened  between  the  publica- 
tion of  the  Clavis  and  the  Geometrie.  It  took  longer  than 
this  period  for  the  Clavis  to  show  evidence  of  its  influence 
upon  mathematical  books  published  in  England;  it  is 
not  probable  that  abroad  the  contact  was  more  immediate 

^  H.  Bosmans,  S.J.,  "La  premiere  edition  de  la  Clavis  Mathematica 
d'Oughtred.  Son  influence  sur  la  Geometrie  de  Descartes,"  Annales 
de  la  societe  scientifique  de  Bruxelles,  35th  year,  1910-11,  Part  II, 
pp.  24-78. 

2  Ibid.,  p.  78. 


Influence  on  Mathematical  Progress         73 

than  at  home.  Our  study  of  seventeenth-century  algebra 
has  led  us  to  the  conviction  that  Oughtred  deserves  a 
higher  place  in  the  development  of  this  science  than  is 
usually  accorded  to  him;  but  that  it  took  several  decennia 
for  his  influence  fully  to  develop. 

THE   SPREAD   OF   OUGHTRED's   NOTATIONS 

An  idea  of  Oughtred's  influence  upon  mathematical 
thought  and  teaching  can  be  obtained  from  the  spread 
of  his  symbolism.  This  study  indicates  that  the  adoption 
was  not  immediate.  The  earliest  use  that  we  have  been 
able  to  find  of  Oughtred's  notation  for  proportion,  A  .Bw 
CD,  occurs  nineteen  years  after  the  Clavis  mathematicae 
of  1 63 1.  In  1650  John  Kersey  brought  out  in  London  an 
edition  of  Edmund  Wingates'  Arithmetique  made  easie, 
in  which  this  notation  is  used.  After  this  date  publica- 
tions employing  it  became  frequent,  some  of  them  being 
the  productions  of  pupils  of  Oughtred.  We  have  seen  it  in 
Vincent  Wing  (1651),^  Seth  Ward  (1653),"  John  Wallis 
(1655), 3  in  ^'R.  B.,"  a  schoolmaster  in  Suffolk,^  Samuel 
Foster  (1659)  ,5  Jonas  Moore  (1660),^  and  Isaac  Barrow 
(1657).'^     In  the  latter  part  of  the  seventeenth  century 

^Vincent  Wing,  Harmonicon  coeleste,  London,  1651,  p.  5. 

^  Seth  Ward,  In  Ismaelis  Bullialdi  astronomiae  philolaicae  funda- 
menta  inqidsitio  brevls,  Oxford,  1653,  p.  7. 

3  John  WalHs,  Elenchus  geometriae  Hobbianae,  Oxford,  1655,  p.  48. 

4  An  Idea  of  Arithnietick,  at  first  designed  for  the  use  of  the  Free 

Schoole  at  Thurlow  in  Suffolk By  R.  B.,  Schoolmaster  there, 

London,  1655,  p.  6. 

s  The  Miscellanies:   or  Mathematical  Lucubrations,  of  Mr.  Samuel 

Foster  ....  by  John  T^vysden,  London,  1659,  p.  i, 

^  Maoris  Arithmetick  in  two  Books,  London,  1660,  p.  89. 

7  Isaac  Barrow,  Euclidis  data,  Cambridge,  1657,  p.  2. 


74  William  Oughtred 

Oughtred's  notation,  A  .B::C .D,  became  the  prevalent, 
though  not  universal,  notation  in  Great  Britain.  A  tre- 
mendous impetus  to  their  adoption  was  given  by  Seth 
Ward,  Isaac  Barrow,  and  particularly  by  John  Wallis,  who 
was  rising  to  international  eminence  as  a  mathematician. 

In  France  we  have  noticed  Oughtred's  notation  for 
proportion  in  Franciscus  Dulaurens  (1667),^  J.  Prestet 
(1675),^  R.  P.  Bernard  Lamy  (i684),3  Ozanam  (i69i),4 
De  I'Hospital  (i696,)s  R.  P.  Petro  Nicolas  (1697). ^ 

In  the  Netherlands  we  have  noticed  it  in  R.  P.  Bernard 
Lamy  (i68o),7  and  in  an  anonymous  work  of  1690.^ 
In  German  and  Italian  works  of  the  seventeenth  cen- 
tury we  have  not  seen  Oughtred's  notation  for  propor- 
tion. 

In  England  a  modified  notation  soon  sprang  up  in 
which  ratio  was  indicated  by  two  dots  instead  of  a  single 
dot,  thus  A:B::C:D.  The  reason  for  the  change  lies 
probably  in  the  inclination  to  use  the  single  dot  to  desig- 
nate decimal  fractions.  W.  W.  Beman  pointed  out  that 
this  modified  symbolism  (:)  for  ratio  is  found  as  early  as 
1657  in  the  end  of  the  trigonometric  and  logarithmic 

^  Francis ci  Dulaurens  Specima  mathematica,  Paris,  1667,  p.  i. 
^  Elemens  des  mathematiques ,  Paris,  1675,  Preface  signed  "J.  P." 

3  Nouveaux  elemens  de  geometric,  Paris,  1692  (permission  to  print 
1684). 

4  Ozanam,  Dictionnaire  mathematique,  Paris,  169 1,  p.  12. 

s  Analyse  des  infiniment  pctits,  Paris,  1696,  p.  11. 

^  Petro  Nicolas,  De  concholdibus  et  cissoidihus  exercitationes 
geometricae,  Toulouse,  1697,  p.  17. 

7  R.  P.  Bernard  Lamy,  Elemens  des  mathematiques,  Amsterdam, 
1692  (permission  to  print  1680). 

*  Nouveaux  elemens  de  geometric,  2d  ed.,  The  Hague,  1690, 
p.  304. 


Influence  on  Mathematical  Progress        75 

tables  that  were  bound  with  Oughtred's  Trigonometria} 
It  is  not  probable,  however,  that  this  notation  was  used 
by  Oughtred  himself.  The  Trigonometria  proper  has 
Oughtred's  A.BwC.D  throughout.  Moreover,  in  the 
English  edition  of  this  trigonometry,  which  appeared  the 
same  year,  1657,  but  subsequent  to  the  Latin  edition,  the 
passages  which  contained  the  colon  as  the  symbol  for 
ratio,  when  not  omitted,  are  recast,  and  the  regular 
Oughtredian  notation  is  introduced.  In  Oughtred's 
posthumous  work,  Opuscula  mathematica  hactenus  inedita, 
1677,  the  colon  appears  quite  often  but  is  most  likely  due 
to  the  editor  of  the  book. 

We  have  noticed  that  the  notation  A:B::C:D  ante- 
dates the  year  1657.  Vincent  Wing,  the  astronomer, 
published  in  165 1  in  London  the  Harmonicon  coeleste,  in 
which  is  found  not  only  Oughtred's  notation  A.BwC.D 
but  also  the  modified  form  of  it  given  above.  The  two 
are  used  interchangeably.  His  later  works,  the  Logistica 
astronomica  (1656),  Doctrina  spherica  (1655),  and  Doctrina 
theorica,  published  in  one  volume  in  London,  all  use  the 
symbols  A:B::C:D  exclusively.  The  author  of  a  book 
entitled.  An  Idea  of  Arithmetick  at  first  designed  for  the 
use  of  the  Free  Schoole  at  Thurlow  in  Suffolk  .  ...  by 
R.  B.,  Schoolmaster  there,  London,  1655,  writes  A:a::C:c, 
though  part  of  the  time  he  uses  Oughtred's  unmodified 
notation. 

We  can  best  indicate  the  trend  in  England  by  indicating 
the  authors  of  the  seventeenth  century  whom  we  have 
found  using  the  notation  A:B::C:D  and  the  authors  of 
the  eighteenth  century  whom  we  have  found  using  A.B:: 
CD.    The  former  notation  was  the  less  common  during 

*  W.  W.  Beman  in  Uintermediaire  des  mathematiciens ,  Paris,  Vol. 
IX,  1902,  p.  229,  question  2424. 


76  William  Oughtred 

the  seventeenth  but  the  more  common  during  the  eight- 
eenth century.  We  have  observed  the  symbols  A\B:: 
C:D  (besides  the  authors  already  named)  in  John  CoUins 
(1659)/  James  Gregory  (1663)/  Christopher  Wren  (1668- 
69) ,3  William  Leybourn  (1673), ^  William  Sanders  (1686) ,5 
John  Hawkins  (1684),^  Joseph  Raphson  (iGgj),''  E.  Wells 
(1698),^  and  John  Ward  (i698).9 

Of  English  eighteenth-century  authors  the  following 
still  clung  to  the  notation  A.B::C.D:  John  Harris' 
translation  of  F.  Ignatius  Gaston  Pardies  (1701)/°  George 
Shelley  (1704),"  Sam  Cobb  (1709)/^  J.  Collins  in  Com- 
mercium  Epistolicum   (17 12),   John   Craig    (1718)/^   Jo. 

*  John  Collins,  The  Mariner^ s  Plain  Scale  New  Plained,  London, 
1659,  p.  25. 

2  James  Gregory,  Optica  promota,  London,  1663,  pp.  19,  48. 

3  Philosophical  Transactions,  Vol.  Ill,  London,  p.  868. 

4  William  Leybourn,  The  Line  of  Proportion,  London,  1673,  p.  14. 

5  Elementa  geometriae  ....  a  Gulielmo  Sanders,  Glasgow,  1686, 
P-  3. 

^  Cocke/s  Decimal  Arithmetick,  ....  perused  by  John  Hawkins, 
London,  1695  (preface  dated  1684),  p.  41. 

7  Joseph  Raphson,  Analysis  Aequationum  universalis,  London, 

1697,  p.  26. 

^  E.  Wells,  Elementa  arithmeticae  numerosae  et  speciosae,  Oxford 

1698,  p.  107. 

9  John  Ward,  A  Compefidium  of  Algebra,  2d  ed.,  London,  1698, 
p.  62. 

^*»  Plain  Elements  of  Geometry  and  Plain  Trigonometry,  London, 
1701,  p.  63. 

"  George  Shelley,  Wingate's  Arithmetick,  London,  1704,  p.  343. 

^^  A  Synopsis  of  Algebra,  Being  a  posthumous  work  of  John  Alex- 
ander of  Bern,  Swisserland.  ....  Done  from  the  Latin  by  Sam. 
Cobb,  London,  1709,  p.  16. 

^3  John  Craig,  De  Calculo  fluent ium,  London,  1718,  p.  35,  The 
notation  A : B :  iC-.D  is  given  also. 


Influence  on  Mathematical  Progress        77 

Wilson  (1724)/  The  latest  use  oi  A.BwC .D  which  has 
come  to  our  notice  is  in  the  translation  of  the  Analytical 
Institutions  of  Maria  G.  Agnesi,  made  by  John  Colson 
sometime  before  1760,  but  which  was  not  published  until 
1 80 1.  During  the  seventeenth  century  the  notation 
A'.B\\C\D  acquired  almost  complete  ascendancy  in 
England. 

In  France  Oughtred's  unmodified  notation  A.BwC.D, 
having  been  adopted  later,  was  also  discarded  later  than 
in  England.  An  approximate  idea  of  the  situation  appears 
from  the  following  data.  The  notation  A.BwC.D  was 
used  by  M.  Carre  (1700),^  M.  Guisnee  (1705),^  M.  de 
Fontenelle  (1727),'*  M.  Varignon  (1725),^  M.  Robillard 
(1753),^  M.  Sebastien  le  Clerc  (1764)/  Clairaut  (1731)/ 
M.  L'Hospital  (i78i).9 

In  Italy  Oughtred's  modified  notation  a,  b::c,  d  was 
used  by  Maria  G.  Agnesi  in  her  Instituzioni  analitiche, 

^  Trigonometry,  2d  ed.,  Edinburgh,  1724,  p.  11. 
2  Methode  pour  la  mesure  des  surfaces,  la  dimension  des  solides 
....  par  M.  Carre  de  Vacademie  r.  des  sciences,  1700,  p.  59, 
^Application  de  Valgehre  a  geometric  ....  Paris,  1705. 

4  Elemens  de  la  geometric  de  Vinfini,  by  M.  de  Fontenelle,  Paris, 
1727,  p.  no. 

5  Eclaircissemens  sur  Vanalyse  des  infiniment  petits,  by  M.  Varig- 
non, Paris,  1725,  p.  87. 

^  Application  de  la  geometric  ordinaire  et  des  calculs  differentiel  et 
integral,  by  M.  Robillard,  Paris,  1753. 

7  Traite  de  geometric  theorique  et  pratique,  new  ed.,  Paris,  1764, 
p.  15- 

*  Recherches  sur  Ics  courhes  a  double  courbure,  Paris,  1731,  p.  13. 

9  Analyse  des  infiniment  petits,  by  the  Marquis  de  L'Hospital. 
New  ed.  by  M.  Le  Fevre,  Paris,  1 781,  p.  41.  In  this  volume  passages 
in  fine  print,  probably  supplied  by  the  editor,  contain  the  notation 
a:b::c:d;   the  parts  in  large  type  give  Oughtred's  original  notation. 


78  William  Oughtred 

Milano,  1748.  The  notation  a:b::c:d  found  entrance 
the  latter  part  of  the  eighteenth  century.  In  Germany 
the  symbolism  a:b  =  c:d,  suggested  by  Leibniz,  found 
wider  acceptance.^ 

It  is  evident  from  the  data  presented  that  Oughtred 
proposed  his  notation  for  ratio  and  proportion  at  a  time 
when  the  need  of  a  specific  notation  began  to  be  generally 
felt,  that  his  symbol  for  ratio  a .  b  was  temporarily  adopted 
in  England  and  France  but  gave  way  in  the  eighteenth 
century  to  the  symbol  a:b,  that  Oughtred's  symbol  for 
proportion  : :  found  almost  universal  adoption  in  England 
and  France  and  was  widely  used  in  Italy,  the  Netherlands, 
the  United  States,  and  to  some  extent  in  Germany;  it  has 
survived  to  the  present  time  but  is  now  being  gradually 
displaced  by  the  sign  of  equality  = . 

Oughtred's  notation  to  express  aggregation  of  terms 
has  received  little  attention  from  historians  but  is  never- 

^  The  tendency  during  the  eighteenth  century  is  shown  in  part 
by  the  following  data:  Jacohi  BernouUi Opera,  Tomus primus,  Geneva, 
1744,  gives  B .A::D .C  on^.  368,  the  paper  having  been  first  pub- 
lished in  1688;  on  p.  419  is  given  GE:AG=LA  -.ML,  the  paper  having 
been  first  published  in  1689.  Bernhardi  Nieuwentiit,  Consider ationes 
circa  analyseos  ad  quantitates  infinite  parvas  applicatae  principia, 
Amsterdam,  1694,  p.  20,  and  Analysis  infinitorum,  Amsterdam, 
1695,  on  p.  276,  have  x:c::s:r.  Paul  Halcken's  Deliciae  mathe- 
maticae,  Hamburg,  1719,  gives  a:b::c:d.  Johannis  Baptistae 
CaraccioH,  Geometria  algehraica  universa,  Rome,  1759,  p.  79,  has 
a.b::c.d.  Delle  corde  ouverto  fibre  elastiche  schediasmi  fisico- 
matematici  del  conte  Giordano  Riccati,  Bologna,  1767,  p.  65,  gives  P:& 
::r:ds.  "Produzioni  ?nathejnatiche"  del  Conte  Giulio  Carlo  de 
Fagnano,  Vol.  I,  Pesario,  1 750,  p.  193,  has  a.b::c.d.  L.  Mascheroni, 
Geojnetrie  du  compas,  translated  by  A.  M.  Carette,  Paris,  1798,  p. 
188,  gives  y's'-2:  :\/2:Lp.  Danielis  Melandri  and  PauUi  Frisi, 
De  theoria  lunae  commentarii,  Parma,  1769,  p.  13,  has  a:b:  :c:d. 
Vicentio  Riccato  and  Hieronymo  Saladino,  Institutiones  analyticae, 
Vol.  I,  Bologna,  1765,  p.  47,  gives  x:a:  :m:n-{-7n.     R.  G.  Boscovich, 


Influence  on  Mathematical  Progress         79 

theless  interesting.  His  books,  as  well  as  those  of  John 
Wallis,  are  full  of  parentheses  but  they  are  not  used  as 
symbols  of  aggregation  in  algebra;  they  are  simply  marks 
of  punctuation  for  parenthetical  clauses.  We  have  seen 
that  Oughtred  writes  {a-\-hy  and  V a-\rh  thus,  Q\a-{-h:^ 
\/:a-\-h:,  or  Q:a-\-h,  \/\a-\-h,  using  on  rarer  occasions 
a  single  dot  in  place  of  the  colon.  This  notation  did  not 
originate  with  Oughtred,  but,  in  slightly  modified  form, 
occurs  in  writings  from  the  Netherlands.  In  1603  C. 
Dibvadii  in  geometriam  Evclidis  demonstratio  numeralis, 
Leyden,  contains  many  expressions  of  this  sort,  1/^-136+ 
1/2048,  signifying  1/(136+]  '2048).  The  dot  is  used  to 
indicate  that  the  root  of  the  binomial  (not  of  136  alone)  is 
called  for.  This  notation  is  used  extensively  in  Ludolphi 
a  Cevlen  de  circulo,  Leyden,  16 19,  and  in  Willebrordi 
Snellii  De  circuli  dimensioned  Leyden,  162 1.  In  place 
of  the  single  dot  Oughtred  used  the  colon  (:),  probably 

Opera  pertinentia  ad  opticam  et  astronomiam,  Bassanl,  1785,  p.  409 
uses  a:b:  -.cd.  Jacob  Bernoulli,  Ars  Conjectandi,  Basel,  1713,  has 
n—r.n  —  i\:c.d.  Pavlini  Chelvicii,  Institiitiones  analyticae,  editio 
post  tertiam  Romanam  prima  in  Ger mania,  Vienna,  1761,  p.  2,  fl.&: : 
c.d.  Christiani  Wolfii,  Elementa  matheseos  universae,  Vol.  Ill, 
Geneva,  1735,  p.  63,  has  AB:AE=i:q.  Johann  Bernoulli,  Opera 
omnia,  Vol,  I,  Lausanne  and  Geneva,  1742,  p.  43,  has  a:b  =  c:d. 
D,  C.  Walmesley,  Analyse  des  mesures  des  rapports  et  des  angles, 
Paris,  1749,  uses  extensively  a.b::c.d,  later  a:b::c:d.  G.  W. 
Krafft,  Institutiones  geometriae  sublimoris,  Tubingen,  1753,  p.  194, 
has  a:b  =  c:d.  J.  H.  Lambert,  Photometria,  1760,  p.  104,  has  C:7r  = 
BC^iMH^.  Meccanica  sublime  del  Dott.  Domenico  Bartaloni,  Naples, 
1765,  has  a:b::c:d.  Occasionally  ratio  is  not  designated  by  a.b, 
nor  by  a:b,  but  by  a,  b,  as  for  instance  in  A.  de  Moivre's  Doctritie 
of  Chance,  London,  1756,  p.  34,  where  he  writes  a,b:  :i,  q.  A  further 
variation  in  the  designation  of  ratio  is  found  in  James  Atkinson's 
Epitome  of  the  Art  of  Navigation,  London,  1718,  p.  24,  namely, 
3. . 2: : 72.  .48.  Curious  notations  are  given  in  Rich.  Balam's 
Algebra,  London,  1653. 


8o  William  Oughtred 

to  avoid  confusion  with  his  notation  for  ratio.  To  avoid 
further  possibihty  of  uncertainty  he  usually  placed  the 
colon  both  before  and  after  the  algebraic  expression  under 
aggregation.  This  notation  was  adopted  by  John  Wallis 
and  Isaac  Barrow.  It  is  found  in  the  writings  of  Des- 
cartes. Together  with  Vieta's  horizontal  bar,  placed 
over  two  or  more  terms,  it  constituted  the  means  used 
almost  universally  for  denoting  aggregation  of  terms  in 
algebra.  Before  Oughtred  the  use  of  parentheses  had 
been  suggested  by  Clavius^  and  Girard.^  The  latter 
wrote,  for  instance,  1/(2+1/3).  While  parentheses  never 
became  popular  in  algebra  before  the  time  of  Leibniz 
and  the  Bernoullis  they  were  by  no  means  lost  sight  of. 
We  are  able  to  point  to  the  following  authors  who  made 
use  of  them:  I.  Errard  de  Bar-le-Duc  (i6i9),3  Jacobo 
de  Billy  (1643),''  one  of  whose  books  containing  this 
notation  was  translated  into  English,  and  also  the  post- 
humous works  of  Samuel  Foster.^  J.  W.  L.  Glaisher 
points  out  that  parentheses  were  used  by  Norwood  in  his 
Trigonometrie  (163 1),  p.  30.^ 

^  Chr.  Clavii  Operum  mathematicorum  tomns  secundus,  Mayence, 
161 1,  Algebra,  p.  39. 

2  Invention  nouvelle  en  Valgebre,  by  Albert  Girard,  Amsterdam, 
1629,  p.  17. 

3  La  geometrie  et  pratique  generate  dHcelle,  par  I.  Errard  de  Bar-le- 
Duc,  Ingenieur  ordinaire  de  sa  Majeste,  3d  ed.,  revised  by  D.  H.  P. 
E.  M.,  Paris,  1619,  p.  216. 

4  Novae  geometriae  clavis  algebra,  authore  P.  Jacobo  de  Billy, 
Paris,  1643,  p.  157;  also  an  Abridgement  of  the  Precepts  of  Algebra. 
Written  in  French  by  James  de  Billy,  London,  1659,  p.  346. 

s  Miscellanies:  or  Mathematical  Lucubrations,  of  Mr.  Samuel 
Foster,  Sometime  publike  Professor  of  Astronomic  in  Gresham  Colledge 
in  London,  London,  1659,  p.  7. 

^Quarterly  Jour,  of  Pure  and  Applied  Math.,  Vol.  XLVI  (London, 
1915),  p.  191. 


Influence  on  Mathematical  Progress         8i 

The  symbol  for  the  arithmetical  difference  between 
two  numbers,  ^,  is  usually  attributed  to  John  Wallis, 
but  it  occurs  in  Oughtred's  Clavis  mathematicae  of  1652, 
in  the  tract  on  Elementi  decimi  Euclidis  declaratio,  at  an 
earlier  date  than  in  any  of  Wallis'  books.  As  Wallis 
assisted  in  putting  this  edition  through  the  press  it  is 
possible,  though  not  probable,  that  the  symbol  was  inserted 
by  him.  Were  the  symbol  Wallis',  Oughtred  would 
doubtless  have  referred  to  its  origin  in  the  preface.  Dur- 
ing the  eighteenth  century  the  symbol  found  its  way  into 
foreign  texts  even  in  far-off  Italy.^  It  is  one  of  three 
symbols  presumably  invented  by  Oughtred  and  which  are 
still  used  at  the  present  time.     The  others  are  X  and  : : . 

The  curious  and  ill-chosen  symbols,  n~  for  ''greater 
than,"  and  _J  for  "less  than,"  were  certain  to  succumb  in 
their  struggle  for  existence  against  Harriot's  admirably 
chosen  >  and  < .  Yet  such  was  the  reputation  of  Oughtred 
that  his  symbols  were  used  in  England  quite  extensively 
during  the  seventeenth  and  the  beginning  of  the  eighteenth 
century.  Considerable  confusion  has  existed  among  alge- 
braists and  also  among  historians  as  to  what  Oughtred's 
symbols  really  were.  Particularly  is  this  true  of  the  sign  for 
"less  than"  which  is  frequently  written ~n.  Oughtred's 
symbols,  or  these  symbols  turned  about  in  some  way,  have 
been  used  by  Seth  Ward,^  John  Wallis,^  Isaac  Barrow,'* 

^  Pietro  Cossali,  Origlne,  trasporto  in  Italia  primi  progressi  in 
essa  deir  algebra,  Vol.  I,  Parmense,  1797,  p.  52. 

2  In  Is.  Bidlialdi  astronomiae  philolaicae  fundamenta  inquisitio 
brevis,  Auctore  Setho  Wardo,  Oxford,  1653,  p.  i. 

3  John  Wallis,  Algebra,  London,  1685,  p.  321,  and  in  some  of  his 
other  works.    He  makes  greater  use  of  Harriot's  symbols. 

^Euclidis  data,  1657,  p.  i;  also  Euclidis  elementorum  libris  XV, 
London,  1659,  p.  i. 


82  William  Oughtred 

John  Kersey/  E.  Wells,^  John  Hawkins,^  Tho.  Baker/ 
Richard  Sault/  Richard  RawHnson/  Franciscus  Dulau- 
rens/  James  Milnes/  George  Cheyne,^  John  Craig /°  Jo. 
Wilson,"  and  J.  Colhns.^^^ 

General  acceptance  has  been  accorded  to  Oughtred's 
symbol  X.  The  first  printed  appearance  of  this  s^mibol 
for  multipUcation  in  1618  in  the  form  of  the  letter  x  hardly 
explains  its  real  origin.  The  author  of  the  ''Appendix" 
(be  he  Oughtred  or  someone  else)  may  not  have  used  the 
letter  x  at  all,  but  may  have  written  the  cross  X,  called 
the  St.  Andrew's  cross,  while  the  printer,  in  the  absence 
of  any  type  accurately  representing  that  cross,  may  have 
substituted  the  letter  x  in  its  place.  The  hypothesis 
that  the  symbol  X  of  multiplication  owes  its  origin  to 
the  old  habit  of  using  directed  bars  to  indicate  that  two 

^  John  Kersey,  Algebra,  London,  1673,  p.  321. 

2  E.  Wells,  Elementa  arithmeticae  niimerosae  et  speciosae,  Oxford, 
1698,  p.  142. 

3  Cocker's  Decimal  Arithmetick,  perused  by  John  Hawkins, 
London,  1695  (preface  dated  1684),  p.  278. 

4  Th.  Baker,  The  Geometrical  Key,  London,  1684,  p.  15. 

5  Richard  Sault,  A  New  Treatise  of  Algebra,  London  (no  date). 

^  Richard  Rawlinson  in  a  pamphlet  without  date,  issued  some- 
time between  1655  and  1668,  containing  trigonometric  formulas. 
There  is  a  copy  in  the  British  Museum. 

'  F.  Dulaurens,  Specima  mathematica.  Pars,  1667,  p.  i. 

^  J.  Milnes,  Sectionum  Gonicarum  elementa,  Oxford,  1702,  p.  42. 

9  Cheyne,  Philosophical  Principles  of  Natural  Religion,  London, 
1705,  P-  55- 

"J.  Craig,  De  calcido  fluentium,  London,  17 18,  p.  86. 

"Jo.  Wilson,  Trigonometry,  2d  ed,,  Edinburgh,  1724,  p.  v. 

^2  Commercium  Epistolicum,  171 2,  p.  20. 


Influence  on  Mathematical  Progress         83 

numbers  are  to  be  combined,  as  for  instance  in  the  multi- 
plication of  23  and  34,  thus, 


has  been  advanced  by  two  writers,  C.  Le  Paige^  and 
Gravelaar.2  Bosmans  is  more  inclined  to  the  belief  that 
Oughtred  adopted  the  symbol  somewhat  arbitrarily, 
much  as  he  did  the  numerous  S3rmbols  in  his  Elementi 
decimi  Euclidis  declaratio.^ 

Le  Paige's  and  Gravelaar's  theory  finds  some  support 
in  the  fact  that  the  cross  X,  without  the  two  additional 
vertical  lines  shown  above,  occurs  in  a  commentary 
published  by  Oswald  Schreshensuchs^  in  1551,  where  the 
sign  is  written  between  two  factors  placed  one  above  the 
other. 

^  C.  Le  Paige,  "Sur  rorigine  de  certains  signes  d'operation," 
Annates  de  la  societe  scientifique  de  Bmxelles,  i6th  year,  1891-92, 
Part  II,  pp.  79-82, 

2  Gravelaar,  "Over  den  oorsprong  van  ons  maalteeken  (X)," 
Wiskundig  Tijdschrift,  6th  year.  We  have  not  had  access  to  this 
article. 

5  H.  Bosmans,  op.  cit,,  p.  40. 

^  Claudii  Ptolemaei  ....  annotationes,  Bale,  1551.  This  refer- 
ence is  taken  from  the  Encyclopedie  des  sciences  mathematiques , 
Tome  I,  Vol.  I,  Fasc.  i,  p.  40. 


CHAPTER  V 

OUGHTRED'S  IDEAS  ON  THE  TEACHING  OF 
MATHEMATICS 

GENERAL   STATEMENT 

Nowhere  has  Oughtred  given  a  full  and  systematic 
exposition  of  his  views  on  mathematical  teaching.  Never- 
theless, he  had  very  pronounced  and  clear-cut  ideas  on  the 
subject.  That  a  man  who  was  not  a  teacher  by  profession 
should  have  mature  views  on  teaching  is  most  interesting. 
We  gather  his  ideas  from  the  quality  of  the  books  he  pub- 
lished, from  his  prefaces,  and  from  passages  in  his  con- 
troversial writing  against  Delamain.  As  we  proceed  to 
give  quotations  unfolding  Oughtred's  views,  we  shall 
observe  that  three  points  receive  special  emphasis:  (i)  an 
appeal  to  the  eye  through  suitable  symbolism;  (2)  em- 
phasis upon  rigorous  thinking;  (3)  the  postponement  of 
the  use  of  mathematical  instruments  until  after  the 
logical  foundations  of  a  subject  have  been  thoroughly 
mastered. 

The  importance  of  these  tenets  is  immensely  reinforced 
by  the  conditions  of  the  hour.  This  voice  from  the  past 
speaks  wisdom  to  specialists  of  today.  Recent  methods 
of  determining  educational  values  and  the  modern  cult 
of  utilitarianism  have  led  some  experts  to  extraordinary 
conclusions.  Laboratory  methods  of  testing,  by  the  nar- 
rowness of  their  range,  often  mislead.  Thus  far  they  have 
been  inferior  to  the  word  of  a  man  of  experience,  insight, 
and  conviction. 

84 


Ideas  on  Teaching  Mathematics  85 

MATHEMATICS,    ''A   SCIENCE   OF   THE  EYE" 

Oughtred  was  a  great  admirer  of  the  Greek  mathe- 
maticians— Eudid,  Archimedes,  Apollonius  of  Perga, 
Diophantus.  But  in  reading  their  works  he  experienced 
keenly  what  many  modern  readers  have  felt,  namely, 
that  the  almost  total  absence  of  mathematical  symbols 
renders  their  writings  unnecessarily  difficult  to  read. 
Statements  that  can  be  compressed  into  a  few  well-chosen 
symbols  which  the  eye  is  able  to  survey  as  a  whole  are 
expressed  in  long-drawn-out  sentences.  A  striking  illus- 
tration of  the  importance  of  symbolism  is  afforded  by  the 
history  of  the  formula 

ix  =  log(cos  x-\-i  sin  x). 

It  was  given  in  Roger  Cotes'  Harmonia  mensuraruniy 
1722,  not  in  symbols,  but  expressed  in  rhetorical  form, 
destitute  of  special  aids  to  the  eye.  The  result  was  that 
the  theorem  remained  in  the  book  undetected  for  185 
years  and  was  meanwhile  rediscovered  by  others.  Owing 
to  the  prominence  of  Cotes  as  a  mathematician  it  is  very 
improbable  that  such  a  thing  could  have  happened  had  the 
theorem  been  thrust  into  view  by  the  aid  of  mathematical 
symbols. 

In  studying  the  ancient  authors  Oughtred  is  reported 
to  have  written  down  on  the  margin  of  the  printed  page 
some  of  the  theorems  and  their  proofs,  expressed  in  the 
symbolic  language  of  algebra. 

In  the  preface  of  his  Clavis  of  163 1  and  of  1647  he  says: 

Wherefore,  that  I  might  more  clearly  behold  the  things 
themselves,  I  uncasing  the  Propositions  and  Demonstrations 
out  of  their  covert  of  words,  designed  them  in  notes  and  species 
appearing  to  the  very  eye.    After  that  by  comparing  the  divers 


S6  William  Oughtred 

affections  of  Theorems,  inequality,  proportion,  affinity,  and 
dependence,  I  tryed  to  educe  new  out  of  them. 

It  was  this  motive  which  led  him  to  introduce  the  many 
abbreviations  in  algebra  and  trigonometry  to  which 
reference  has  been  made  in  previous  pages.  The  peda- 
gogical experience  of  recent  centuries  has  indorsed  Ought- 
red's  view,  provided  of  course  that  the  pupil  is  carefully 
taught  the  exact  meaning  of  the  symbols.  There  have 
been  and  there  still  are  those  who  oppose  the  intensive  use 
of  symbolism.  In  our  day  the  new  symbolism  for  all 
mathematics,  suggested  by  the  school  of  Peano  in  Italy, 
can  hardly  be  said  to  be  received  with  enthusiasm.  In 
Oughtred's  day  symbolism  was  not  yet  the  fashion.  To 
be  convinced  of  this  fact  one  need  only  open  a  book  of 
Edmund  Gunter,  with  whom  Oughtred  came  in  contact 
in  his  youth,  or  consult  the  Principia  of  Sir  Isaac  Newton, 
who  flourished  after  Oughtred.  The  mathematical  works 
of  Gunter  and  Newton,  particularly  the  former,  are 
surprisingly  destitute  of  mathematical  symbols.  The 
philosopher  Hobbes,  in  a  controversy  with  John  WaUis, 
criticized  the  latter  for  that  ''Scab  of  Symbols,"  where- 
upon Wallis  replied: 

I  wonder  how  you  durst  touch  M.  Oughtred  for  fear  of  catch- 
ing the  Scab.     For,  doubtlesse,  his  book  is  as  much  covered 

over  with  the  Scab  of  Symbols,  as  any  of  mine As  for 

my  Treatise  of  Conick  Sections,  you  say,  it  is  covered  over  with 
the  Scab  of  Symbols,  that  you  had  not  the  patience  to  examine 
whether  it  is  well  or  ill  demonstrated.^ 

^  Due  Carrection  for  Mr.  Hobbes.  Or  Schoole  Discipline,  for  not 
saying  his  Lessons  right.  In  answer  to  his  Six  Lessons,  directed  to  the 
Professors  of  Mathematicks.  By  the  Professor  of  Geometry.  Oxford, 
1656,  pp.  7,  47,  50. 


Ideas  on  Teaching  Mathematics  87 

Oughtred  maintained  his  view  of  the  importance  of 
symbols  on  many  different  occasions.  Thus,  in  his  Circles 
of  Pro  portion  y  1632,  p.  20: 

This  manner  of  setting  downe  Theoremes,  whether  they  be 
Proportions,  or  Equations,  by  Symboles  or  notes  of  words,  is 
most  excellent,  artificiall,  and  doctrinall.  Wherefore  I  earn- 
estly exhort  every  one,  that  desireth  though  but  to  looke  into 
these  noble  Sciences  Mathematicall,  to  accustome  themselves 
unto  it:  and  indeede  it  is  easie,  being  most  agreeable  to  reason, 
yea  even  to  sence.  And  out  of  this  working  may  many  singular 
consectaries  be  drawne:  which  without  this  would,  it  may  be, 
for  ever  lye  hid. 

RIGOROUS   THINKING  AND  THE  USE   OF  INSTRUMENTS 

The  author's  elevated  concept  of  mathematical  study 
as  conducive  to  rigorous  thinking  shines  through  the  fol- 
lowing extract  from  his  preface  to  the  1647  Clavis: 

....  Which  Treatise  being  not  written  in  the  usuall  syn- 
thetical manner,  nor  with  verbous  expressions,  but  in  the  inven- 
tive way  of  Analitice,  and  with  symboles  or  notes  of  things 
instead  of  words,  seemed  imto  many  very  hard;  though  indeed 
it  was  but  their  owne  diffidence,  being  scared  by  the  newnesse 
of  the  dehvery;  and  not  any  difficulty  in  the  thing  it  selfe. 
For  this  specious  and  symbohcall  manner,  neither  racketh  the 
memory  with  multipUcity  of  words,  nor  chargeth  the  phantasie 
with  comparing  and  laying  things  together;  but  plainly  pre- 
senteth  to  the  eye  the  whole  course  and  processe  of  every  opera- 
tion and  argumentation. 

Now  my  scope  and  intent  in  the  first  Edition  of  that  my 
Key  was,  and  in  this  New  FiUng,  or  rather  forging  of  it,  is,  to 
reach  out  to  the  ingenious  lovers  of  these  Sciences,  as  it  were 
Ariadnes  thread,  to  guide  them  through  the  intricate  Labyrinth 
of  these  studies,  and  to  direct  them  for  the  more  easie  and  full 
understanding   of   the   best   and   antientest   Authors.  .-.  .  . 


S8  William  Oughtred 

That  they  may  not  only  learn  their  propositions,  which  is  the 
highest  point  of  Art  that  most  Students  aime  at;  but  also  may 
perceive  with  what  solertiousnesse,  by  what  engines  of  aequa- 
tions,  Interpretations,  Comparations,  Reductions,  and  Dis- 
quisitions, those  antient  Worthies  have  beautified,  enlarged, 

and  first  found  out  this  most  excellent  Science Lastly, 

by  framing  like  questions  problematically,  and  in  a  way  of 
Analysis,  as  if  they  were  already  done,  resolving  them  into  their 
principles,  I  sought  out  reasons  and  means  whereby  they  might 
be  effected.  And  by  this  course  of  practice,  not  without  long 
time,  and  much  industry,  I  found  out  this  way  for  the  helpe 
and  facilitation  of  Art. 

Still  greater  emphasis  upon  rigorous  thinking  in  mathe- 
matics is  laid  in  the  preface  to  the  Circles  of  Proportion 
and  in  some  parts  of  his  Apologeticall  Epistle  against 
Delamain.  In  that  preface  William  Forster  quotes  the 
reply  of  Oughtred  to  the  question  how  he  (Oughtred)  had 
for  so  many  years  concealed  his  invention  of  the  slide 
rule  from  himself  (Forster)  whom  he  had  taught  so  many 
other  things.    The  reply  was: 

That  the  true  way  of  Art  is  not  by  Instruments,  but  by 
Demonstration:  and  that  it  is  a  preposterous  course  of  vulgar 
Teachers,  to  begin  with  Instruments,  and  not  with  the  Sciences, 
and  so  in-stead  of  Artists,  to  make  their  Scholers  only  doers 
of  tricks,  and  as  it  were  luglers:  to  the  despite  of  Art,  losse 
of  previous  time,  and  betraying  of  willing  and  industrious 
wits,  vnto  ignorance,  and  idlenesse.  That  the  vse  of  Instru- 
ments is  indeed  excellent,  if  a  man  be  an  Artist:  but  contemp- 
tible, being  set  and  opposed  to  Art.  And  lastly,  that  he  meant 
to  commend  to  me,  the  skill  of  Instruments,  but  first  he  would 
haue  me  well  instructed  in  the  Sciences." 

Delamain  took  a  different  view,  arguing  that  instru- 
ments might  very  well  be  placed  in  the  hands  of  pupils 
from  the  start.    At  the  time  of  this  controversy  Delamain 


Ideas  on  Teaching  Mathematics  89 

supported  himself  by  teaching  mathematics  in  London 
and  he  advertised  his  abiUty  to  give  instruction  in  mathe- 
matics, including  the  use  of  instruments.  Delamain 
brought  the  charge  against  Oughtred  of  unjustly  calling 
"many  of  the  [British]  Nobility  and  Gentry  doers  of  trickes 
and  juglers."    To  this  Oughtred  replies: 

As  I  did  to  Delamain  and  to  some  others,  so  I  did  to 
WUliam  Forster:  I  freely  gave  him  my  helpe  and  instruction  in 
these  faculties:  only  this  was  the  difference,  I  had  the  very 
first  moulding  (as  I  may  say)  of  this  latter:  But  Delamain 
was  already  corrupted  with  doring  upon  Instruments,  and  quite 
lost  from  ever  being  made  an  Artist:  I  suffered  not  WiUiam 
Forster  for  some  time  so  much  as  speake  of  any  Instrument, 
except  only  the  Globe  it  selfe;  and  to  explicate,  and  worke 
the  questions  of  the  Sphaere,  by  the  way  of  the  Analemma: 
which  also  himselfe  did  describe  for  the  present  occasion.  And 
this  my  restraint  from  such  pleasing  avocations,  and  holding 
him  to  the  strictnesse  of  percept,  brought  forth  this  fruit,  that 
in  short  time,  even  by  his  owne  skill,  he  could  not  onely  use 
any  Instrument  he  should  see,  but  also  was  able  to  delineate  the 
like,  and  devise  others.^ 

As  representing  Delamain's  views,  we  make  the  fol- 
lowing selection  from  his  Grammelogia  (London,  about 
1633),  the  part  near  the  end  of  the  book  and  bearing  the 
title,  "In  the  behalf e  of  vulgar  Teachers  and  others," 
where  Delamain  refers  to  Oughtred's  charge  that  the 
scholars  of  "vulgar"  teachers  are  "doers  of  tricks,  as  it 
were  iuglers."    Delamain  says: 

....  Which  words  are  neither  cautelous,  nor  subterfugious, 
but  are  as  downe  right  in  their  plainnesse,  as  they  are  touching, 
and  pernitious,  by  two  much  derogating  from  many,  and  glan- 
cing upon  many  noble  personages,  with  too  grosse,  if  not  too 
base  an  attribute,  in  tearming  them  doers  of  tricks,  as  it  were  to 

*  Oughtred,  Apologeticall  Epistle,  p.  27. 


90  William  Oughtred 

iuggle:  because  they  perhaps  make  use  of  a  necessitie  in  the 
furnishing  of  themselves  with  such  knowledge  by  Practicall 
Instrumentall  operation,  when  their  more  weighty  negotiations 
will  not  permit  them  for  Theoreticall  figurative  demonstration; 
those  that  are  guilty  of  the  aspertion,  and  are  touched  therewith 
may  answer  for  themselves,  and  studie  to  be  more  Theoreticall, 
than  Practicall:  for  the  Theory,  is  as  the  Mother  that  produceth 
the  daughter,  the  very  sinewes  and  life  of  Practise,  the  excel- 
lencie  and  highest  degree  of  true  Mathematicall  Knowledge: 
but  for  those  that  would  make  but  a  step  as  it  were  into  that 
kind  of  Learning,  whose  onely  desire  is  expedition,  and  faciUtie, 
both  which  by  the  generall  consent  of  all  are  best  effected  with 
Instrument,  rather  then  with  tedious  regular  demonstrations,  it 
was  ill  to  checke  them  so  grosly,  not  onely  in  what  they  have 
Practised,  but  abridging  them  also  of  their  liberties  with  what 
they  may  Practise,  which  aspertion  may  not  easily  be  slighted 
off  by  any  glosse  or  Apologie,  without  an  Ingenuous  confession, 
or  some  mentaU  reservation :  To  which  vilification,  howsoever, 
in  the  behalf e  of  my  selfe,  and  others,  I  answer;  That  Instru- 
mentall operation  is  not  only  the  Compendiating,  and  facihtat- 
ing  of  Art,  but  even  the  glory  of  it,  whole  demonstration  both 
of  the  making,  and  operation  is  soly  in  the  science,  and  to  an 
Artist  or  disputant  proper  to  be  knowne,  and  so  to  all,  who 
would  truly  know  the  cause  of  the  Mathematicall  operations 
in  their  originall;  But,  for  none  to  know  the  use  of  a  Mathe- 
maticall Instrumen[t],  except  he  knowes  the  cause  of  its  opera- 
tion, is  somewhat  too  strict,  which  would  keepe  many  from 
affecting  the  Art,  which  of  themselves  are  ready  enough  every 
where,  to  conceive  more  harshly  of  the  difficultie,  and  impos- 
sibilitie  of  attayning  any  skill  therein,  then  it  deserves,  because 
they  see  nothing  but  obscure  propositions,  and  perplex  and 
intricate  demonstrations  before  their  eyes,  whose  unsavoury 
tartnes,  to  an  imexperienced  palate  like  bitter  pills  is  sweetned 
over,  and  made  pleasant  with  an  Instrumentall  compendious 
facilitie,  and  made  to  goe  downe  the  more  readily,  and  yet  to 
retaine  the  same  vertue,  and  working;  And  me  thinkes  in  this 


Ideas  on  Teaching  Mathematics  91 

queasy  age,  all  helpes  may  bee  used  to  procure  a  stomacke,  all 
hates  and  invitations  to  the  declining  studie  of  so  noble  a  Science, 
rather  then  by  rigid  Method  and  generall  Lawes  to  scarre  men 
away.  All  are  not  of  like  disposition,  neither  all  (as  was  sayd 
before)  propose  the  same  end,  some  resolve  to  wade,  others 
to  put  a  finger  in  onely,  or  wet  a  hand:  now  thus  to  tye  them 
to  an  obscure  and  Theoricall  forme  of  teaching,  is  to  crop  their 

hope,  even  in  the  very  bud The  beginning  of  a  mans 

knowledge  even  in  the  use  of  an  Instrument,  is  first  founded  on 
doctrinal  precepts,  and  these  precepts  may  be  conceived  all 
along  in  its  use :  and  are  so  f arre  from  being  excluded,  that  they 
doe  necessarily  concomitate  and  are  contained  therein:  the 
practicke  being  better  understood  by  the  doctrinall  part,  and 
this  later  explained  by  the  Instrumentall,  making  precepts 
obvious  unto  sense,  and  the  Theory  going  along  with  the 
Instrument,  better  informing  and  inlightning  the  understanding, 
etc.  vis  vnita  fortior,  so  as  if  that  in  Phylosophy  bee  true,  Nihil 
est  [in]  intellectu  quod  non  prius  fuit  in  sensu. 

The  difference  between  Oughtred  and  Delamain  as  to 
the  use  of  mathematical  instruments  raises  important 
questions.  Should  the  slide  rule  be  placed  in  the  hands  of 
a  boy  before,  or  after,  he  has  mastered  the  theory  of  loga- 
rithms ?  Should  logarithmic  tables  be  withheld  from  him 
until  the  theoretical  foundation  is  laid  in  the  mind  of  the 
pupil?  Is  it  a  good  thing  to  let  a  boy  use  a  surveying 
instrument  unless  he  first  learns  trigonometry?  Is  it 
advisable  to  permit  a  boy  to  familiarize  himself  with  the 
running  of  a  dynamo  before  he  has  mastered  the  under- 
lying principles  of  electricity?  Does  the  use  of  instru- 
ments ordinarily  discourage  a  boy  from  mastery  of  the 
theory  ?  Or  does  such  manipulation  constitute  a  natural 
and  pleasing  approach  to  the  abstract  ?  On  this  particu- 
lar point,  who  showed  the  profounder  psychological  in- 
sight, Oughtred  or  Delamain  ? 


92  William  Oughtred 

In  July,  19 14,  there  was  held  in  Edinburgh  a  celebra- 
tion of  the  three-hundredth  anniversary  of  the  invention 
of  logarithms.  On  that  occasion  there  was  collected  at 
Edinburgh  university  one  of  the  largest  exhibits  ever  seen 
of  modern  instruments  of  calculation.  The  opinion  was 
expressed  by  an  experienced  teacher  that  "weapons  as 
those  exhibited  there  are  for  men  and  not  for  boys,  and 
such  danger  as  there  may  be  in  them  is  of  the  same 
character  as  any  form  of  too  early  specialization." 

It  is  somewhat  of  a  paradox  that  Oughtred,  who  in  his 
student  days  and  during  his  active  years  felt  himself 
impelled  to  invent  sun-dial?,  planispheres,  and  various 
types  of  slide  rules — instruments  which  represent  the 
most  original  contributions  which  he  handed  down  to 
posterity — should  discourage  the  use  of  such  instruments 
in  teaching  mathematics  to  beginners.  That  without  the 
aid  of  instruments  he  himself  should  have  succeeded  so 
well  in  attracting  and  inspiring  young  men  constitutes 
the  strongest  evidence  of  his  transcendent  teaching  ability. 
It  may  be  argued  that  his  pedagogic  dogma,  otherwise 
so  excellent,  here  goes  contrary  to  the  course  he  himself 
followed  instinctively  in  his  self-education  along  mathe- 
matical lines.  We  read  that  Sir  Isaac  Newton,  as  a  child, 
constructed  sun-dials,  windmills,  kites,  paper  lanterns, 
and  a  wooden  clock.  Should  these  activities  have  been 
suppressed  ?  Ordinary  children  are  simply  Isaac  Newtons 
on  a  smaller  intellectual  scale.  Should  their  activities 
along  these  lines  be  encouraged  or  checked  ? 

On  the  other  hand,  it  may  be  argued  that  the  paradox 
alluded  to  above  admits  of  explanation,  like  all  paradoxes, 
and  that  there  is  no  inconsistency  between  Oughtred's 
pedagogic  views  and  his  own  course  of  development.  If 
he  invented  sun-dials,  he  must  have  had  a  comprehension 


Ideas  on  Teaching  Mathematics  93 

of  the  cosmic  motions  involved;  if  he  solved  spherical 
triangles  graphically  by  the  aid  of  the  planisphere,  he  must 
have  understood  the  geometry  of  the  sphere,  so  far  as  it 
relates  to  such  triangles;  if  he  invented  slide  rules,  he 
had  beforehand  a  thorough  grasp  of  logarithms.  The 
question  at  issue  does  not  involve  so  much  the  invention 
of  instruments,  as  the  use  by  the  pupil  of  instruments 
already  constructed,  before  he  fully  understands  the 
theory  which  is  involved.  Nor  does  Sir  Isaac  Newton's 
activity  as  a  child  establish  Delamain's  contention.  Of 
course,  a  child  should  not  be  discouraged  from  manual 
activity  along  the  line  of  producing  interesting  toys  in 
imitation  of  structures  and  machines  that  he  sees,  but 
to  introduce  him  to  the  realm  of  abstract  thought  by  the 
aid  of  instruments  is  a  different  proposition,  fraught  with 
danger.  A  boy  may  learn  to  use  a  slide  rule  mechanically 
and,  because  of  his  abiUty  to  obtain  practical  results, 
feel  justified  in  foregoing  the  mastery  of  underlying  theory; 
or  he  may  consider  the  ability  of  manipulating  a  surveying 
instrument  quite  sufiicient,  even  though  he  be  ignorant  of 
geometry  and  trigonometry;  or  he  may  learn  how  to 
operate  a  dynamo  and  an  electric  switchboard  and  be 
altogether  satisfied,  though  having  no  grasp  of  electrical 
science.  Thus  instruments  draw  a  youth  aside  from  the 
path  leading  to  real  intellectual  attainments  and  real 
efficiency;  they  allure  him  into  lanes  which  are  often 
blind  alleys.     Such  were  the  views  of  Oughtred. 

Who  was  right,  Oughtred  or  Delamain?  It  may  be 
claimed  that  there  is  a  middle  ground  which  more  nearly 
represents  the  ideal  procedure  in  teaching.  Shall  the  slide 
rule  be  placed  in  the  student's  hands  at  the  time  when 
he  is  engaged  in  the  mastery  of  principles  ?  Shall  there 
be  an  alternate  study  of  the  theory  of  logarithms  and  of 


94  William  Oughtred 

the  slide  rule — on  the  idea  of  one  hand  washing  the  other — 
until  a  mastery  of  both  the  theory  and  the  use  of  the 
instrument  has  been  attained?  Does  this  method  not 
produce  the  best  and  most  lasting  results?  Is  not  this 
Delamain's  actual  contention  ?  We  leave  it  to  the  reader 
to  settle  these  matters  from  his  own  observation,  knowl- 
edge, and  experience. 

Newton's  comments  on  oughtred 

Oughtred  is  an  author  who  has  been  found  to  be  of 
increasing  interest  to  modern  historians  of  mathematics. 
But  no  modern  writer  has,  to  our  knowledge,  pointed  out 
his  importance  in  the  history  of  the  teaching  of  mathe- 
matics. Yet  his  importance  as  a  teacher  did  receive 
recognition  in  the  seventeenth  century  by  no  less  distin- 
guished a  scientist  than  Sir  Isaac  Newton.  On  May  25, 
1694,  Sir  Isaac  Newton  wrote  a  long  letter  in  reply  to  a 
request  for  his  recommendation  on  a  proposed  new  course 
of  study  in  mathematics  at  Christ's  Hospital.  Toward 
the  close  of  his  letter,  Newton  says : 

And  now  I  have  told  you  my  opinion  in  these  things,  I  wiU 
give  you  Mr.  Oughtred's,  a  Man  whose  judgment  (if  any  man's) 
may  be  safely  relyed  upon.  For  he  in  his  book  of  the  circles 
of  proposition,  in  the  end  of  what  he  writes  about  Navigation 
(page  184)  has  this  exhortation  to  Seamen.  "And  if,"  saith 
he,  "the  Masters  of  Ships  and  Pilots  will  take  the  pains  in  the 
Journals  of  their  Voyages  dihgently  and  faithfully  to  set  down 
in  severaU  columns,  not  onely  the  Rumb  they  goe  on  and  the 
measure  of  the  Ships  way  in  degrees,  and  the  observation  of 
Latitude  and  variation  of  their  compass;  but  alsoe  their  con- 
jectures and  reason  of  their  correction  they  make  of  the  aber- 
rations they  shall  find,  and  the  qualities  and  condition  of  their 
ship,  and  the  diversities  and  seasons  of  the  winds,  and  the 
secret  motions  or  agitations  of  the  Seas,  when  they  begin,  and 


Ideas  on  Teaching  Mathematics  95 

how  long  they  continue,  how  farr  they  extend  and  with  what 
inequaUty;  and  what  else  they  shall  observe  at  Sea  worthy 
consideration,  and  will  be  pleased  freely  to  communicate  the 
same  with  Artists,  such  as  are  indeed  skilfuU  in  the  Mathe- 
maticks  and  lovers  and  enquirers  of  the  truth:  I  doubt  not 
but  that  there  shall  be  in  convenient  time,  brought  to  light 
many  necessary  precepts  which  may  tend  to  y®  perfecting  of 
Navigation,  and  the  help  and  safety  of  such  whose  Vocations 
doe  inforce  them  to  commit  their  lives  and  estates  in  the  vast 
Ocean  to  the  providence  of  God."  Thus  farr  that  very  good 
and  judicious  man  Mr.  Oughtred.  I  will  add,  that  if  instead  of 
sending  the  Observations  of  Seamen  to  able  Mathematicians 
at  Land,  the  Land  would  send  able  Mathematicians  to  Sea, 
it  would  signify  much  more  to  the  improvem*  of  Navigation  and 
safety  of  Mens  lives  and  estates  on  that  element.^ 

May  Oughtred  prove  as  instructive  to  the  modern 
reader  as  he  did  to  Newton! 

^  J.  Edleston,  Correspondence  of  Sir  Isaac  Newton  and  Professor 
Cotes,  London,  1850,  pp.  279-92. 


INDEX 


Adam,  Charles,  71 

Agnesi,  Maria  G.,  77 

Alexander,  J.,  76 

Allen,  E.,  35 

Analysis,  19,  20 

Apollonius  of  Perga,  20,  85 

Archimedes,  18,  20,  85 

Aristotle,  69 

Ashmole,  E.,  13 

Atkinson,  J.,  79 

Atwood,  56 

Aubrey,  3,  7,  8,  12-16,  58,  59 

Austin,  58 

Baker,  T.,  82 

Balam,  R.,  79 

Bar-le-Duc,  de,  80 

Barrow,  S.,  i,  32,  73,  74,  80,  81 

Bartaloni,  D.,  79 

Beman,  W.  W.,  74,  75 

Bernoulli,  Jakob,  78-80 

Bernoulli,  John,  79,  80 

Billingsley's  Euclid,  15 

Billion,  20 

Billy,  Jacobo  de,  80 

Binomial  formula,  25,  29 

Bliss,  P.,  60 

Boscovich,  R.  G.,  78 

Bosmans,  H.,  72,  83 

Boyle,  R.,  i,  69 

Braunmiihl,  von,  39 

Brearly,  W.,  59 

Briggs,  6,  36,  55 

Brookes,  Christopher,  7,  53,  59 

Cajori,  F.,  27,  39,  40,  47 
Cantor,  M.,  40,  41 
Caraccioli,  J.  B.,  78 
Cardan,  71 
Carre,  77 

Carrete,  N.  M.,  78 
CaryU,  C,  7 


Cataldi,  67 

Cavalieri,  65,  66 

Cavendish,  Charles,  17,  62,  66 

Charles  I,  9,  60 

Chelvicius,  P.,  79 

Cheyne,  G.,  82 

Circles  of  Proportion,  35,  37,  48, 

49,  SI,  59,  87,  88 
Clairaut,  77 
Clark,  A.,  3 
Clark,  G.,  63 
Clarke,  F.  L.,  3 
Clavis  mathematicae,  i,  5,  10,  14, 

17-35,  45,  46,  51,  57-63,  68- 

73,  81,  85,  87 
Clavius,  26,  80 
Clerc,  le,  77 
Cobb,  S.,  76 
Cocker,  76,  82 
Collins,  John,  15,  19,  63,  64,  67, 

68,  76,  82 
Colson,  J.,  77 
Conchoid,  12 
Conic  sections,  11,  53 
Cossali,  P.,  81 
Cotes,  R.,  I,  85 
Craig,  J.,  76,  82 
Cross,  symbol  of  multiplication, 

27,  38,  55,  56,  82,  83 
Cubic  equations,  28,  34,  42,  45 

Decimal  fractions,  notation  of,  21 
Degree,  centesimal  division,  39 
Delamain,  R.,  4,  9, 10, 11,  47, 48, 

51,60,84,88,89,91,93,94 
De  Moivre,  32,  79 
De  Morgan,  A.,  5,  16,  37,  46,  47, 

54 
Descartes,  R.,  i,  25,  47,  57,  68- 

72,  80 
Dibuadius,  79 
Difference,  symbol  for,  27,  81 


97 


98 


William  Oughtred 


Diophantus,  63,  85 

Division,  abbreviated,  21,  23,  24 

Dulaurens,  F.,  74,  82 

Earl  of  Arundel,  10,  13,  15,  17 

Edleston,  J.,  95 

Enestrom,  G.,  40 

Equations,   solution  of,   18,  28, 

29,  31,  34,  39-45,  87 
Errard  de  Bar-le-Duc,  80 
Eton  College,  3,  4 
Euclid,  I,  15,  18,  20,  25,  27,  28, 

79,  81,  83,  85 
Euler,  L.,  37,  39 
Ewart,  59 
Exponents,  25,  28,  29 

Fagnano,  de,  78 
Flower,  56 
FonteneUe,  de,  77 
Forster,  W.,  35,  48,  59,  88 
Foster,  S.,  27,  67,  69,  73,  80,  89 
Frisi,  P.,  78 

Gascoigne,  59,  61 

Gauss,  C.  F.,  48 

Geysius,  67 

Ghetaldi,  70,  71 

Gibson,  67 

Girard,  A.,  32,  80 

Glaisher,  J.  W.  L.,  54-56,  64,  80 

Glorioso,  67,  68 

Grammelogia,  4,  47,  89 

Gravelaar,  83 

Greater  than,  symbol  for,  81 

Greatrex,  R.,  15 

Gregory,  D.,  32 

Gregory,  J.,  27,  76 

Gresham  College,  i,  6,  27,  59,  61, 

80 
Guisnee,  77 
Gunter,  E.,  37,  47,  86 
Gunter's  scale,  37 

Halcken,  P.,  78 
Hales,  J.,  7 
Halley,  E.,  i,  18,  69 
Hankel,  H.,  40 
Harper,  T.,  18 


Harriot,  T.,  45,  47,  57,  58,  69- 

71,81 
Harris,  J,,  76 
Hartlib,  69 
Haughton,  A.,  35,  59 
Hawkins,  J.,  76,  82 
Hearn,  56 
Helmholtz,  48 
Henry,  J/,  48 
Henry  van  Etten,  52,  53 
Henshaw,  T.,  8,  58,  61 
Hobbes,  73,  86 
Hollar,  14 
Holsatus,  13 
Hooganhuysen,  64 
Hooke,  Rb.,  i 
Horner's  method,  45 
Horology,  18,  50 
Horrox,  J.,  4 
Hospital,  de  1',  74,  77 
Howard,      Th.     See      Earl  of 

Arundel. 
Howard,  W.,  17,  18,  59 
Hutchinson,  A.,  6 

Invisible  college,  i 

Joule,  48 

Kepler,  J.,  6 

Kersey,  J.,  32,  73,  82 

Keylway,  R.,  65 

King,  67 

Kings  College,  Cambridge,  3,  35 

Krafft,  G.  W.,  79 

Lambert,  J.  H.,  79 

Lamy,  R.  P.  B.,  74 

Laud,  Archbishop,  65 

Leake,  W.,  53 

Le  Clerc,  77 

Leech,  W.,  59 

Le  Fevre,  77 

Leibniz,  47,  78,  80 

Leonelli,  56 

Le  Paige,  de,  83 

Less  than,  symbol  for,  81 

Leurechon,  52 

Leybourn,  35,  64,  76 


Index 


99 


Lichfield,  Mrs.,  19 

LiUy,  W.,  8,  9 

Locke,  J.,  67 

Logarithms,  6,  21,  27,  28,  38,  39, 

42,    46,    54-56,   65,    92,    93; 

natural,  55;    radix  method  of 

computing,  55,  5^ 
Lower,  W.,  58 
Ludolph  a  Ceulen,  79 

Manning,  56 
Manning,  O.,  7,  8,  13-15 
Mascheroni,  L.,  78 
Mayer,  R.,  47 
Melandri,  D.,  78 
Mercator,  N.,  13 
Mersenne,  63 
Milboum,  W.,  45 
Million,  20 
Milnes,  J.,  82 
Moivre,  de,  32,  79 
Moore,  Jonas,  32,  54,  58,  73 
Moreland,  S.,  70 
Morse,  R.,  48 

Multiplication,  abbreviated,  21, 
22,  24;  symbol  for,  27,  82,  83 
Mydorge,  54 

Napier,  J.,  6,  7,  21,  27,  38,  39, 

52,  54,  57,  59 
Napier's  analogies,  39 
Newton,  Sir  Isaac,  i,  25,  29,  40, 

41,  45,47,  59,65,  86,  92-95 
Nichols,  J.,  6,  14 
Nicolas,  R.  P.  P.,  74 
Nieuwentiit,  B.,  78 
Norwood,  R.,  37,  38,  8° 

Opiiscula    mathematica    hactenus 

inedita,  16,  21,  75 
Orchard,  56 

Oughtredus  explicatus,  64 
Ozanam,  74 

IT,  symbol  for,  32 

Paige,  C.  dej  83 
Pardies,  76 

Parentheses,  26,  79,  80 
Partridge,  S.,  47 


Peano,  86 

Perfect  number,  41 

Pitiscus,  15 

Planisphere,  53,  92,  93 

Prestet,  J.,  74 

Price,  II 

Proportion,  notation  for,  26,  27, 

73-79 
Protheroe,  58 
Ptolemy,  83 

Quadratic  equation,  29,  31,  34 

Radix  method,  55,  56 

Rahn,  27 

Raphson,  J.,  40,  41,  76 

Ratio,  notation  of,  21,  73-80 

Rawlinson,  R.,  39,  82 

Regula  falsa,  18 

Regular  solids,  18 

Riccati,  G.,  78 

Riccati,  V.,  78 

Rigaud,  7,  12,  13,  19,  48,  61-66, 

68 
Robillard,  77 

Robinson,  W.,  13,  48,  59,  62,  63 
Rooke,  L.,  59,  61 

Saladini,  H.,  78 

Sanders,  W.,  76 

Sault,  R.,  82 

Scarborough,    Charles,    16,    54, 

58,60 
Schooten,  Van,  i 
Schreshensuchs,  O.,  83 
Scratch  method,  23 
Shakespeare,  52 
Shelley,  G.,  76 
Shipley,  A.  E.,  i 
Shuttleworth,  59 
Slide  rule,  9,  46-49,  5°,  60,  88,  93 
Smethwyck,  58 
Smith,  J.,  50 
SneUius,  W.,  79 
Solids,  regular,  18 
Speidell,  John,  38,  55 
Spherical  triangles,  53,  54,  93 
Stokes,  R.,  35,  36,  58 
Sudell,  59 
Sun  dials,  5,  9,  50,  51,  52,  60,  92 


lOO 


William  Oughtred 


Tannery,  P.,  71 
Todhunter,  60 
Torporley,  58 

Triangles,  spherical,  53,  54,  93 
Trlgonometria,  21,  36,  55,  75 
Trigonometric    functions,    sym- 
bols for,  36,  37,  55,  56 
Trigonometrie,  21,  35,  39 
Trisection  of  angles,  28 
Twysden,  59,  68,  69,  73 

Varignon,  77 

Vieta,  I,  2,  25,  32,  33,  35,  39- 

41,  45,  63,  67,  70,  71 
Vlack,  65 
Von  Braimmiihl,  39 

Wadham  College,  5,  53 
Wallis,  John,  i,  19,  27,  33,  45, 
57-59,  63,  64,  66-74,  79-81,  86 


Walmesley,  D.  C,  79 

Ward,  Bishop,  13 

Ward,  John,  76 

Ward,  Seth,  55,  58,  60,  68,  73, 

74,  81 
Watch-making,  18,  50 
Weber,  W.  E.,  48 
Weddle,  56 
Wells,  E.,  76,  82 
Wharton,  60 
Whitlock,  B.,  8,  9 
Wilson,  J.,  77,  82 
Wing,  v.,  73,  75 
Wingate,  E.,  32,  47,  73 
Wolf,  Christian,  79 
Wood,  A.,  60,  61 
Wood,  R.,  18,  59 
Wren,  Christopher,  5,  58,  59,  76 
Wright,  E.,  6,  27,  38,  54 
Wright,  S.,  54 


Date  Due 

1 

i 
1 

. 

! 

! 
1 

1 

^ 

BOSTON  dOLLEGE 


3  9031   01 


546708  7 


BOSTON  COllIfitSClENCEUIMf 


iOS(^'^      math/dept. 


BOSTON  COLLEGE  LIBRARY 

UNIVERSITY  HEIGHTS 
CHESTNUT  HILL,  MASS. 


Books  may  be  kept  for  two  weeks  and  may  be 
renewed   for    the    same   period,  unless    reserved. 

Two  cents  a  day  is  charged  for  each  book  kept 
overtime. 

If  you    cannot   find   what   you   want,  ask    the 
Librarian  who  will  be  glad  to  help  you. 

The  borrower  is  responsible  for  books  drawn 
on  his  card  and  for  all  fines  accruing  on  the  same. 


11  m 


i 


I' 


■■■i 


II  ! 


!i 


lii'lHitMj'llllilil 


ili!'i!:'  ( 


iili   III!  Hi 


